additive combinatorics and its applications in theoretical ...€¦ · introduction additive...
TRANSCRIPT
Additive Combinatorics and its Applications inTheoretical Computer Science (draft)
Shachar LovettUniversity of California, San DiegoComputer Science and Engineering
September 30, 2013
Introduction
Additive combinatorics is a branch of mathematics which lies at the intersection of combina-torics, number theory, Fourier analysis and ergodic theory. It studies approximate notionsof algebraic structures, such as vector spaces or finite fields. For example, let A be a subsetof an Abelian group G. Define A + A = a + a′ : a, a′ ∈ A to be the set of sums of pairsof elements from A. It is relatively easy to see that |A + A| ≥ |A| and that equality holdsonly if A is a subgroup of G, or a coset of a subgroup. A classic question studied in additivecombinatorics is what is the structure of A if A+A is not much larger than A. In particular,one would like to show that in such a case, A is close in a certain sense to a subgroup. Moregenerally, problems in additive combinatorics study the behaviour of sets under addition andmultiplication, and its relation to structural properties of the sets.
In recent years, several connections between additive combinatorics and theoreticalcomputer science have been discovered. Techniques and results in additive combinatoricswere applied to problems in coding theory, property testing, hardness of approximation,computational complexity, communication complexity, randomness extraction and pseudo-randomness. In general, these advances occurred because additive combinatorics providednew tools to study the structures arising in various computational settings.
The goal of this survey is to provide an introduction to additive combinatorics for stu-dents and researchers in theoretical computer science. This is the main driving force behindthe choice of topics covered, as well as the applications of them which are presented. By far,this is not a comprehensive survey. We chose to present proofs of the more basic results inorder to give the flavour of the different techniques, and to refer to the original papers for themore advanced proofs. There are several other very useful resources on the topic: the book”Additive Combinatorics” by Tao and Vu [56] gives a detailed description of many results inadditive combinatorics and their applications, mainly in number theory; A mini-course onadditive combinatorics by Barak et al [10] explores these ideas and some of their applicationsin computer science; the survey ”Selected Results in Additive Combinatorics: An Exposi-tion” by Viola [57] covers the basic results related to set addition and their application inSamorodnitsky’s theorem [47] on linearity testing for functions with multiple outputs; andthe survey ”The polynomial Freiman-Ruzsa conjecture” by Green [24] explores a centralconjecture in this area. A more specialized survey by the author [34] gives an exposition ofa recent important result by Sanders [51] which gets close to proving this conjecture.
In this current survey, we focus on three main topics: the structure of set addition,arithmetic progressions in sets, and the sum-product phenomena. For each, we cover basicresults with detailed proofs, state more advanced results, and give several applications incomputer science. Notably, an area which is not covered in this survey is higher-orderFourier analysis. This is an emerging field which greatly generalizes the classic theory ofFourier analysis, and which has deep connections to additive combinatorics. However, inorder to keep this survey concise, we decided to defer an exposition of higher-order Fourieranalysis and its applications to a future survey.
2
Chapter 1
Set addition
Let G be an Abelian group, where a motivating example to have in mind is G = Fn (i.e. avector space over a field). Let A be a finite subset of the group. The sumset of A is definedto be the set
2A = A+ A = a+ a′ : a, a′ ∈ A
It is simple to see that |2A| ≥ |A|. Equality holds only if A is a subgroup of G, or a coset ofa subgroup: we can assume 0 ∈ A by shifting A (that is, replace A with A− a0 = a− a0 :a ∈ A for some a0 ∈ A. This does not change the size of A or 2A). Now, since 0 ∈ A wehave A ⊆ 2A. Moreover, since by assumption |2A| = |A| then in fact 2A = A. That is, A isclosed under addition, and hence is a subgroup of G.
The focus of this chapter is on relaxations of this simple claim. We will consider variousnotions of approximate subgroups and how they relate to each other.
1.1 Ruzsa calculus
Ruzsa calculus is a general method to argue about sumsets of several sets and how theyrelate to each other. We will show a simple example of its applicability. Let A,B be twosubsets of an Abelian group G we define A + B = a + b : a ∈ A, b ∈ B and A − B =a − b : a ∈ A, b ∈ B to be their sumset and difference set, respectively. The followingtriangle inequality is both simple and useful.
Claim 1.1.1 (Ruzsa triangle inequality). Let A,B,C be subsets of an Abelian group. Then
|A||B − C| ≤ |A+B||A+ C|.
Proof. Any element b−c ∈ B−C can be written in |A| distinct ways as b−c = (a+b)−(a+c).Formally, define a map f : A× (B−C)→ (A+B)× (A+C) as follows: for any x ∈ B−Cfix a representation x = b− c with b ∈ B, c ∈ C, and map f(a, x) = (a+ b, a+ c). This mapis injective, hence |A||B − C| ≤ |A+B||A+ C|.
We can use Claim 1.1.1 to define a distance between sets.
3
Definition 1.1.2 (Ruzsa distance). The Ruzsa distance between sets A,B is defined as
d(A,B) = log|A−B||A|1/2|B|1/2
.
Claim 1.1.3. The Ruzsa distance is a distance function, that is, it is symmetric and obeysthe triangle inequality.
Proof. Symmetry holds since |B−A| = |A−B|. We need to prove that d(A,C) ≤ d(A,B)+d(B,C), which is equivalent to
|A− C||B| ≤ |A−B||B − C|.
This follows from Claim 1.1.1 applied to −B,A,C.
Ruzsa distance can be used to show that various notions of ”bounded growth” are relatedto each other. For example, the following claim shows that if |A+A| ≤ K|A| then |A−A| ≤K2|A|. We will see more general forms of these relations later.
Corollary 1.1.4. Let A,B be subsets of an Abelian group with |A − B| ≤ K|A|1/2|B|1/2.Then |A− A| ≤ K2|A|.
Proof.
|A− A||A|
= exp(d(A,A)) ≤ exp(d(A,B) + d(B,A)) =|A−B|2
|A||B|≤ K2.
1.2 The span of sets of small doubling
Let A be a subset of an Abelian group G. The doubling constant of A is K = |A + A|/|A|;although sometimes it will be convenient to look on |A−A|/|A|. We would like to understandthe structure of sets of small doubling. We already saw that K = 1 corresponds to subgroupsof G. Hence, we might expect that small values of K correspond to sets which are close insome sense to subgroups. The first question we would like to understand is what is the sizeof the smallest subgroup containing A. For example, when G = Fn is a vector space this isequivalent to the size of the linear subspace spanned by A.
We start by an argument of Laba [32] showing that when |A−A| < (3/2)|A| the set A isclose to a subgroup in a very strong sense: A− A is already a subgroup. The constant 3/2is tight, as can be seen by taking A = 0, 1 ⊂ Z.
Lemma 1.2.1 ( [32]). Let A ⊂ G be a set such that |A − A| < (3/2)|A|. Then A − A is asubgroup of G.
4
Proof. We will show that for any x ∈ A− A we have
|A ∩ (A+ x)| > |A|/2.
Then, for any x, y ∈ A − A we have that (A + x) ∩ (A + y) is nonempty. Hence, there area1, a2 ∈ A such that a1 +x = a2 + y. This means that x− y = a2−a1 ∈ A−A. Thus, A−Ais closed under differentiation, and hence must be a subgroup. Now, let x = a − a′ wherea, a′ ∈ A. Then
|A ∩ (A+ x)| = |(A− a) ∩ (A− a′)|.
Since A − a,A − a′ are both subsets of A − A of size |A − a| = |A − a′| = |A|, we have bythe inclusion-exclusion principle that
|(A− a) ∩ (A− a′)| ≥ |A− a|+ |A− a′| − |A− A| > |A|/2.
The structure of A with general bounded doubling is unfortunately more complicated.In the special case of G = Rn Freiman [22] showed that sets of small doubling must alsohave small dimensions by a relatively simple argument, based on the convexity of Euclideanspace. We will later see analogs of this result over other general Abelian groups, where theproofs are more complex.
Lemma 1.2.2 ( [22]). Let A ⊂ Rn with |A+A| ≤ K|A|. Then A lies in an affine subspaceof dimension at most 2K.
The dimension in the lemma is tight up to lower order terms, as can be seen by choosingA to be a set of 2K linearly independent vectors.
Proof. We will prove the following: if A has affine dimension d then
|A+ A| ≥ (d+ 1)|A| −(d+ 1
2
).
Since |A| ≥ d this implies that if |A+A| ≤ K|A| then d ≤ 2K − 1. We will prove the claimby induction on d and |A|. Let M(A) = (a + a′)/2 : a, a′ ∈ A be the set of ”mid-points”of A. Clearly, |M(A)| = |A+A|. Let C(A) be the convex hull of the points on A, where byassumption it is a d-dimensional polytope. Let a∗ ∈ A be an arbitrary vertex of C(A), andlet A′ = A \ a∗. There are two cases to consider:
(i) The affine dimension of A′ is d − 1. In this case, the mid-points (a + a∗)/2 for a ∈ Aare outside C(A′). Hence,
|M(A)| ≥ |A|+ |M(A′)| ≥ |A|+ d|A′| −(d
2
)= (d+ 1)|A| −
(d+ 1
2
).
5
(ii) The affine dimension of A′ is d. Let a1, . . . , at ∈ A′ be the neighboring vertices to a∗ inC(A), where t ≥ d. The points a∗ and (a∗ + ai)/2 for 1 ≤ i ≤ t are mid-points outsideC(A′). Hence,
|M(A)| ≥ (t+ 1) + |M(A′)| ≥ d+ 1 + (d+ 1)|A′| −(d+ 1
2
)= (d+ 1)|A| −
(d+ 1
2
).
Ruzsa [46] established an analog of Freiman’s theorem for arbitrary Abelian groups. Inthe following, a group G has torsion r ≥ 1 if r · g = 0 for all g ∈ G.
Theorem 1.2.3 ( [46]). Let G be an Abelian group of torsion r. Let A ⊂ G be a subset with|A + A| ≤ K|A|. Then there exists a subgroup H < G of size |H| ≤ K2rK
4 |A| such thatA ⊂ H.
Before proceeding to prove Theorem 1.2.3, let us first consider the parameters in it.Consider the following example: let G = Fnp , a vector space over a finite field Fp, whichhas torsion r = p. Let U, V ⊂ Fnp be two subspaces with U ∩ V = 0; and set A =U×v1, . . . , v2K where v1, . . . , v2K ∈ V are linearly independent. Then, A+A = U×vi+vj :1 ≤ i ≤ j ≤ 2K and |A+A|/|A| ≈ K. However, the size of the minimal subspace containingA is p2K |U | = (p2K/2K)|A|. Thus, an exponential dependency on K is unavoidable in therelation between the doubling constant and the size of the minimal subgroup containing thewhole set. We will later see a conjecture (the Polynomial Freiman-Ruzsa conjecture) whichspeculates that a polynomial dependency on K is possible if we allow the subspace to containnot all A, but a noticeable fraction if it. This polynomial dependency seems to be crucialfor most of the applications in computer science.
In the current settings, the above example led to the following conjecture of Ruzsa onthe optimal parameters for his theorem.
Conjecture 1.2.4 ( [46]). There exists an absolute constant C ≥ 2 such that the followingholds. For any Abelian group G of torsion r, and any subset A ⊂ G with |A + A| ≤ K|A|,there exists a subgroup H < G of size |H| ≤ rCK |A| such that A ⊂ H.
A sequence of works [19, 21, 26, 31, 48] established this conjecture for r = 2 with C = 2,and more generally for any prime torsion [35] also with C = 2. In fact, the bounds obtainedexactly match the example above of K linearly independent vectors.
Theorem 1.2.5 ( [35]). Let p be prime. Let A ⊂ Fnp be a subset with |A+A| ≤ K|A|. Then
there exists a subspace H ⊂ Fnp such that A ⊂ H and |H| ≤ p2K−2
2K−1|A|.
We will focus here on the proof of Theorem 1.2.3, as it is more general and allows us tointroduce another important theorem: sets of small doubling don’t grow too much even withrepeated additions and subtractions. We will only use the following theorem. We defer itsproof to Section 1.3, where we will prove a more general result. Recall that 2A = A+ A.
6
Theorem 1.2.6. If |A+ A| ≤ K|A| then |2A− 2A| ≤ K4|A|.
Proof of Theorem 1.2.4 assuming Theorem 1.2.6. Let A ⊂ G be such that |2A| ≤ K|A|. Wemay assume 0 ∈ A by possible replacing A with A − a for some a ∈ A. We will bound thesize of the subgroup spanned by A.
Let B = b1, . . . , bm be a maximal collection of elements such that bi ∈ 2A−A and thesets bi − A are all disjoint. Since bi − A ⊂ 2A− 2A we have that
m ≤ |2A− 2A|/|A| ≤ K4.
We first argue that 2A−A ⊆ B+A−A. To see that, choose any x ∈ 2A−A. By constructionthere exists b ∈ B such that (x−A)∩ (b−A) 6= ∅. That is, x−a = b−a′ for some a, a′ ∈ A.Hence x = b+ a− a′ ∈ B + A− A′. As a corollary, we get that for any k ≥ 1,
kA− A ⊆ (k − 1)B + A− A.
The proof is by induction on k:
kA− A = A+ ((k − 1)A− A) ⊂ A+ (k − 2)B + A− A ⊂ (k − 1)B + A− A.
Now, let 〈A〉 be the subgroup of G spanned by A, and 〈B〉 be the subgroup spanned by B.Then
〈A〉 =⋃k≥1
kA ⊂⋃k≥1
(kA− A) ⊂⋃k≥1
(k − 1)B + A− A = 〈B〉+ A− A.
Hence we get that|〈A〉| ≤ |〈B〉| · |A− A| ≤ |〈B〉| ·K2|A|,
where we applied Corollary 1.1.4 to bound |A−A|. To conclude, we need to bound the sizeof 〈B〉. As the group G has torsion r and |B| ≤ K4 we have |〈B〉| ≤ rK
4which concludes
the proof.
1.3 The growth of sets of small doubling
Assume that |A + A| is not much larger than |A|. How large can be |A + A + A|? or|A+A−A−A|? The following theorem of Plunneke [38] which was re-discoverd by Ruzsa [45]gives a nearly complete answer. Define the k-th iterated sumset of A as
kA = a1 + . . .+ ak : a1, . . . , ak ∈ A,
and more generally for k, ` ≥ 1 define
kA− `A = a1 + . . .+ ak − ak+1 − . . .− a` : a1, . . . , ak+` ∈ A.
Theorem 1.3.1 (Plunneke-Ruzsa theorem [38, 45]). Let G be an Abelian group, A,B ⊂ Gbe sets of equal size |A| = |B| such that |A+B| ≤ K|A|. Then |kA− `A| ≤ Kk+`|A|.
7
In particular, we get the following corollary.
Corollary 1.3.2. If |A+ A| ≤ K|A| or |A− A| ≤ K|A| then |kA− `A| ≤ Kk+`|A|.
The original proof of Theorem 1.3.1 was based on graph theory and was quite involved.Recently, a much simpler proof recently discovered by Petridis [37]. We will present hisproof.
Proof of Theorem 1.3.1. Let A,B be sets such that |A| = |B| and |A + B| ≤ K|A|. LetB0 ⊂ B be a nonempty set minimizing the ratio K0 = |A+B0|/|B0|. We have that
• K0 ≤ K.
• |A+B0| = K0|B0|.
• For any subset B′ ⊂ B0, |A+B′| ≥ K0|B′|.
The theorem will following from the following lemma.
Lemma 1.3.3. For any set C ⊂ G,
|A+B0 + C| ≤ K0|B0 + C|.
We first show why the theorem follows. By induction on k we have that |B0 + kA| ≤Kk
0 |B0| ≤ Kk|B0|. By Ruzsa triangle inequality (Claim 1.1.1),
|B0||kA− `A| ≤ |B0 + kA||B0 + `A| ≤ Kk+`|B0|2
and hence|kA− `A| ≤ Kk+`|B0| ≤ Kk+`|A|.
Proof of Lemma 1.3.3. The proof is by induction on |C|. If C = x then the lemma holdsas
|A+B0 + x| = |A+B0| = K0|B0| = K0|B0 + x|.
If |C| > 1 let C = x ∪ C ′. Let B′ ⊂ B0 be defined as
B′ = b ∈ B0 : A+ b+ x ⊂ A+B0 + C ′.
Then, we can decompose A+B0 + C as
A+B0 + C = (A+B0 + C ′) ∪ ((A+B0 + x) \ (A+B′ + x)).
This allows us to bound the size of A+B0 + C inductively by
|A+B0 + C| ≤ |A+B0 + C ′|+ |(A+B0 + x) \ (A+B′ + x)|≤ K0|B0 + C ′|+ |A+B0| − |A+B′|≤ K0(|B0 + C ′|+ |B0| − |B′|),
8
where we used the facts that A + B′ + x ⊂ A + B0 + x and the assumption on B0. Toconclude, we need to show that |B0 + C ′|+ |B0| − |B′| ≤ |B0 + C|. Let B′′ ⊂ B0 be definedas
B′′ = b ∈ B0 : b+ x ∈ B0 + C ′.
Note that B′′ ⊂ B′. We can decompose B0 + C as a disjoint union
B0 + C = (B0 + C ′) ∪ ((B0 + x) \ (B′′ + x)).
Hence,
|B0 + C| = |B0 + C ′|+ |B0| − |B′′| ≥ |B0 + C ′|+ |B0| − |B′|
which is what we needed to show.
1.4 Statistical set addition
Assume that A is a set where a1 + a2 typically lies in a small set, but not always. It mightbe the case that A + A is very large. Consider for example the following example: A ⊂ Fn2is a union of a subspace of dimension n/2 and a random subset of Fn2 of size 2n/2. ThatPra,a′∈A[a+ a′ ∈ A] ≥ 1/4 but typically |A+A| ≈ 2n/2|A|. However, this example motivatesthe following relaxation: if we restrict our attention to a large subset of A, then it does have asmall doubling constant. This turns out to be true in general. The first proof of the generalresult was by Balog and Szemeredi [1]. Later, Gowers [23] found a simpler proof whichsimultaneously obtains much better bounds (improving exponential loss of parameters to apolynomial loss). This theorem is known as the BSG (Balog-Szemeredi-Gowers) theoremand has found a large number of applications, both in mathematics and in computer science.We will present a variant of the proof due to Sudakov, Szemeredi and Vu [53]; the reader isalso referred to an exposition of Viola [57] for a somewhat simplified proof over Fn2 .
Theorem 1.4.1 (BSG Theorem [1, 23, 53]). Let A,B ⊂ G be subsets of an Abelian groupwith |A| = |B| = N . Assume that there exists a set C ⊂ G with |C| = cN such that
Pra∈A,b∈B
[a+ b ∈ C] ≥ ε,
where a ∈ A, b ∈ B are chosen uniformly and independently. Then there exist subsetsA′ ⊂ A,B′ ⊂ B of size |A′|, |B|′ ≥ (ε2/16)N such that |A′ +B′| ≤ 212c3(1/ε)5 ·N .
Note that in particular, we also have |A′ +A′| ≤ poly(c, 1/ε) ·N by Theorem 1.3.1. Theproof of Theorem 1.4.1 is based on the following lemma in graph theory. It states that inany dense graph, there is a large subset of the vertices, such that for any pair of vertices inthis large set there are many short paths between them.
9
Lemma 1.4.2 ( [53]). Let H = (A,B,E) be a bi-partite graph with vertex sets A,B and edgesE ⊂ A× B. Assume that |A| = |B| = N and |E| = εN2. Then there exist A′ ⊂ A,B′ ⊂ Bof size |A′|, |B|′ ≥ (ε2/16)N such that for any a ∈ A′, b ∈ B′ there are at least 2−12ε5N2
paths of length 3 between a and b.
We first prove Theorem 1.4.1 based on Lemma 1.4.2.
Proof of Theorem 1.4.1 assuming Lemma 1.4.2. Let H be the bi-partite graph with vertexsets A,B and edges E = (a, b) : a+b ∈ C. Applying Lemma 1.4.2 to H, let A′ ⊂ A,B′ ⊂ Bbe the guaranteed subsets. We will upper bound |A′ +B′|. For any a ∈ A′, b ∈ B′, considera path (a, b′, a′, b). Clearly,
y = a+ b = (a+ b′)− (b′ + a′) + (a′ + b) = x− x′ + x′′,
where x = a + b, x′ = a′ + b, x′′ = a′ + b are elements of C, since they are all edges of thegraph H. Moreover, any element y ∈ A′ + B′ can be represented as y = x − x′ + x′′ forat least 2−12ε5N2 ordered triples (x, x′, x′′) with x, x′, x′′ ∈ C. On the other hand, C hascardinality cN and hence there at c3N3 such triples in total. Hence,
|A′ +B′| ≤ c3N3
2−12ε5N2= 212c3(1/ε)5N.
We next prove Lemma 1.4.2. The main idea is the following simple a powerful observationcalled dependent random choice. Let v ∈ A be a randomly chosen vertex and let Γ(v) ⊂ B beits set of neighbors. Typically, |Γ(v)| is large since the graph is dense. A pair (u,w) ∈ B iscalled bad if they have a few common neighbors. The main idea is that the set Γ(v) containsonly a small number of bad pairs. Indeed, if (u,w) ∈ Γ(v) is a bad pair, then we must havechosen v from a the small set of common neighbors of u,w.
Fix now a vertex v so that Γ(v) is large and there is only a small fraction of bad pairsin Γ(v). Let B′ ⊂ Γ(v) be the set of vertices which are part of only a few bad pairs. Byassumption, B′ is also big. Let A′ be the set of vertices with many neighbors in B′. Sincethe graph is dense, one can show that A′ is big. Now, fix any a ∈ A′, b ∈ B′. There aremany neighbors b′ of a in B′, and most of them are not in a bad pair with b. Hence, thereare many choices for a′ ∈ A which are common neighbors of b, b′, which means that there aremany paths (a, b′, a′, b) of length three between a and b. We now proceed with the detailedproof.
Proof of Lemma 1.4.2. Let H = (A,B,E) with |A| = |B| = N and |E| = εN2. The averagedegree of a vertex in H is εN . We first delete all vertices from B of degree ≤ (ε/2)N . As wecan delete at most (ε/2)N2 edges this way, the resulting graph has ≥ (ε/2)N2 edges. Forconvenience we keep the same notation. We are working with a graph H = (A,B,E) with|A| = N,N/2 ≤ |B| ≤ N and where any vertex in B has degree at least (ε/2)N .
10
Let v ∈ A be a uniformly chosen vertex, and let Γ(v) ⊂ B be the neighborhood of v.We prove a few properties of Γ(v). Let X = |Γ(v)| be the number of neighbors of v. Theexpected size of X is
E[X] =
∑v∈A |Γ(v)||A|
=|E||A|≥ (ε/2) ·N.
We call a pair (u,w) with u,w ∈ B bad if the number of common neighbors of (u,w) is lessthan (ε3/128) · N . Let Y be the number of bad pairs which belong to Γ(v). For any fixedbad pair (u,w), the probability that u,w ∈ Γ(v) is equal to the probability that v is one ofthe common neighbors of u,w, hence
Pr[u,w ∈ Γ(v)] ≤ ε3/128.
Since |B| ≤ N there are at most(N2
)bad pairs. Thus, by linearity of expectation,
E[Y ] ≤ (ε3/128) ·(N
2
)≤ (ε3/256) ·N2.
Let S ⊂ Γ(v) be the set of vertices u ∈ Γ(v) which form a bad pair with at least (ε2/32) ·Nother elements w ∈ Γ(v). Letting Z = |S| we have
Z · (ε2/32) ≤ 2Y,
where the factor 2 is due to the fact that a bad pair (u,w) can be counted twice. Hence
E[Z] ≤ 64ε2 · E[Y ] ≤ (ε/4) ·N.We set B′ = Γ(v) \ S. By linearity of expectation,
E[B′] ≥ E[X]− E[Z] ≥ (ε/4) ·N.Hence, there is a choice for v ∈ A such that |B′| ≥ (ε/4) · N ; and for any u ∈ B′ there areat most (ε2/32) ·N elements w ∈ B′ such that (u,w) is a bad pair.
Next, we define A′ = a ∈ A : |Γ(a) ∩ B′| ≥ (ε2/16) · N. We first show that A′ is nottoo small. The number of edges between A and B′ is
|E(A,B′)| ≤ N |A′|+ (ε2/16)N · |A \ A′| ≤ N |A′|+ (ε2/16) ·N2.
On the other hand, any element b ∈ B has degree at least (ε/2) ·N , hence
|E(A,B′)| ≥ (ε/2)N · |B′| ≥ (ε2/8) ·N2.
We conclude that|A′| ≥ (ε2/16)N2.
Now we show that for any a ∈ A′, b ∈ B′ there are many paths of length 3 connecting aand b. Indeed, there are at least (ε2/16) ·N neighbors b′ of a in B′. Out of which, at most(ε2/32) ·N form a bad pair with b. Thus, there are (ε2/32)N choices for b′ ∈ Γ(a)∩B′ suchthat (b′, b′) is not a bad pair. That is, there are at least (ε2/128) ·N common neighbors a′ ofb, b′. These defined a path (a, b′, a′, b) between a and b. The number of these pathes is henceat least
(ε2/32)N · (ε3/128)N = (ε5/212) ·N.
11
1.5 Linearity testing
The first application of additive combinatorics that we will see will be in property testing.Property testing asks whether specific properties of large objects can be efficiently detectedby algorithms, which only look on a small part of these objects. It was initiated by the workof Blum, Luby, and Rubinfeld [9] who observed that given a function f : Fn2 → F2, it ispossible to inquire the value of f on a few random points, and accordingly probabilisticallydistinguish between the case that f is a linear function and the case that f has to bemodified on at least a noticeable fraction of points to become a linear function. Inspired bythis observation, Rubinfeld and Sudan [44] defined the concept of property testing which isnow a major area of research in theoretical computer science. Roughly speaking, to test afunction for a property means to examine the value of the function on a few random points,and accordingly (probabilistically) distinguish between the case that the function has theproperty and the case that it is not too close to any function with that property. We willstart by describing the basic BLR linearity test, and its analysis based on Fourier analysis.We will then see how to generalize this to testing linearity for functions f : Fn2 → Fm2 form > 1, where the analysis will use additive combinatorics.
1.5.1 Testing linear functions
It will be instructive to explain in more detail the BLR linearity test. We will focus onboolean functions, although the analysis can be extended to functions f : Fnp → Fp. Letf : Fn2 → F2 be a function. The BLR test is very simple: choose randomly and uniformlyx, y ∈ Fn2 , and accept f if
f(x+ y) = f(x) + f(y).
Clearly, if f is a linear function (e.g. f(x) =∑aixi mod 2) then this probabilistic test
always accepts f . What may be more surprising is that if f is far from all linear functions,then the BLR test rejects f with a noticeable probability. The importance of this result isthat this is achieved while reading only 3 values from the function f .
Let us make a few basic definitions: for functions f, g : Fn2 → F2 their distance is thefraction of elements in which they differ,
dist(f, g) =1
2n
∣∣∣∣x ∈ Fn2 : f(x) 6= g(x)∣∣∣∣.
Let Linearn = g(x) =∑aixi : a ∈ Fn2 be the collection of all linear functions. The distance
of f from linear functions is the minimal fraction of points in f which needs to be changedto get some linear function,
dist(f,Linearn) = ming∈Linearn
dist(f, g)
We now present the quantitative analysis of the BLR test. We follow the analysis of [6]which uses Fourier analysis to obtain better bounds, with a simpler proof. As Fourier analysiswill be useful for us also later, this is time well spent.
12
Theorem 1.5.1 (BLR test analysis [6, 9]). Let f : Fn2 → F2. If f is linear then the BLRtest always accept f . If dist(f,Linearn) ≥ ε then the BLR test rejects f with probability atleast ε.
As promised, the proof of Theorem 1.5.1 uses Fourier analysis. Let F : Fn2 → R be afunction, where we typically would have F (x) = (−1)f(x) for f : Fn2 → F2. The charactersof Fn2 are the set of real-valued functions χα : Fn2 → −1, 1 for α ∈ Fn2 defined as
χα(x) = (−1)〈α,x〉,
where 〈α, x〉 =∑n
i=1 αixi. The following summarizes some of the basic properties of thecharacters.
Fact 1.5.2.
1. The set of characters is an orthogonal basis for functions Fn2 → R under the innerproduct 〈f, g〉 = 2−n
∑x∈Fn
2f(x)g(x). In particular, χ0 ≡ 1 and
∑x∈Fn
2χα(x) = 0 for
all α 6= 0.
2. Additivity: χα(x+ y) = χα(x)χα(y).
3. Multiplicativity: for any α, β ∈ Fn2 , χα+β(x) = χα(x)χβ(x).
4. Any function F : Fn2 → R can be decomposed as a linear combination of the characters,
F (x) =∑α∈Fn
2
F (α)χα(x).
The coefficients F (α) are called the Fourier coefficients of F .
5. The Fourier coefficients can be directly computed as
F (α) = 2−n∑x∈Fn
2
F (x)χα(x).
6. Parseval identity: ∑α∈Fn
2
F (α)2 = 2−n∑x∈Fn
2
F (x)2.
In particular, if F takes values in −1, 1 then∑
α∈Fn2F (α)2 = 1.
Proof of Theorem 1.5.1. Let f : Fn2 → F2. If is obvious that if f is linear then the BLRtest always accepts f . We will show that if dist(f,Linearn) = ε then the test rejects fwith noticable probability. Let F (x) = (−1)f(x). The Fourier coefficients of F measure the
13
distance of f from all linear functions. Specifically, let α ∈ Fn2 and g(x) = 〈α, x〉 be a linearfunction. Then
F (α) = 2−n∑x∈Fn
2
(−1)f(x)+g(x)
= 2−n (|x : f(x) = g(x)| − |x : f(x) 6= g(x)|)= 1− 2−n · 2|x : f(x) 6= g(x)| = 1− 2 · dist(f, g).
In particular, if dist(f, g) ≥ ε then F (α) ≤ 1−2ε. We next use Fourier analysis to analyze theacceptance probability of the BLR test. Recall, the test accepts f if f(x+ y) = f(x) + f(y)where x, y ∈ Fn2 are uniformly chosen. Now,
Ex,y∈Fn2[F (x+ y)F (x)F (y)] = Pr[f(x+ y) = f(x) + f(y)]− Pr[f(x+ y) 6= f(x) + f(y)]
= 2 · Pr[BLR test accepts f ]− 1.
Taking the Fourier expansion of F , we get
E[F (x+ y)F (x)F (y)] = Ex,y∈Fn2
∑α∈Fn
2
F (α)χα(x) ·∑β∈Fn
2
F (β)χβ(x) ·∑γ∈Fn
2
F (γ)χγ(x+ y)
=
∑α,β,γ∈Fn
2
F (α)F (β)F (γ) · Ex,y [χα(x)χβ(y)χγ(x+ y)]
=∑
α,β,γ∈Fn2
F (α)F (β)F (γ) · Ex [χα+γ(x)] · Ey [χβ+γ(y)]
=∑α∈Fn
2
F (α)3,
where the last equality follows because Ex∈Fn2[χα(x)] = 1 if α = 0, and is zero otherwise.
Thus, we can express the acceptance probability of the BLR test as
Pr[BLR test accepts f ] =1
2
(1 +
∑α
F (α)3
).
We next bound the sum∑F (α)3. Since we assume dist(f,Linearn) = ε we have F (α) ≤
1− 2ε. Also, by Parseval inequality,∑F (α)2 = 1. Hence∑
α
F (α)3 ≤ (1− 2ε)∑α
F (α)2 = 1− 2ε,
andPr[BLR test accepts f ] ≤ 1− ε.
14
1.5.2 Testing linear maps
Let us consider now a more general scenario where we have functions f : Fn2 → Fm2 , and wewant to test the distance of f to the set of all linear functions mapping n bits to m bits.This is defined analogously to the previous definition: let Linearn,m denote the set of linearfunctions g : Fn2 → Fm2 . The distance from f to Linearn,m is the fraction of entries of f whichneeds to be changed to make f a linear function.
The same test as before seems reasonable: choose x, y ∈ Fn2 uniformly and accept f iff(x+y) = f(x)+f(y). This test still accepts always linear functions, but it is less clear howto analyze its behaviour on functions which are far from linear. The previous approach usingFourier analysis does not seem to generalize. Instead, we will see that additive combinatoricscan be used to analyze this. The actual reason is that additive combinatorics allows for ageneralization of Fourier analysis, called higher-order Fourier analysis. We will not seethis generalization yet, but it is good to keep this in mind. We will follow the work ofSamorodnitsky [47] who introduced this approach to computer science. For a survey on thisresult see also [57].
Theorem 1.5.3 ( [47]). Let f : Fn2 → Fm2 be a function, and consider the following test:choose x, y ∈ Fn2 uniformly, and accept f if
f(x+ y) = f(x) + f(y).
If f is a linear function then the test always accepts f . If the distance of f from linearfunctions is ε then the test rejects f with probability at least δ(ε) ≥ exp(−εO(1)).
We note that the rejection probability here is worse than that achieved for booleanfunctions in Theorem 1.5.1. In fact, a polynomial dependency between δ and ε is conjectured,and is equivalent to a conjecture we will see soon, the polynomial Freiman-Ruzsa conjecture.
Proof. Lets assume that the test rejects f with a small probability δ, and we will see that fis close to a linear function. Let A = (x, f(x)) : x ∈ Fn2 ⊂ Fn+m
2 be the graph of f . Notethat |A| = 2n. Then, we know that
Pra,b∈A
[a+ b ∈ A] = Prx,y∈Fn
2
[(x+ y, f(x) + f(y)) ∈ A] = Prx,y∈Fn
2
[f(x+ y) = f(x) + f(y)] ≥ δ.
Applying the Balog-Szemeredi-Gowers theorem (Theorem 1.4.1), we obtain a subset A′ ⊂ Aof size |A′| ≥ δO(1) · 2n such that |A′ + A′| ≤ (1/δ)O(1)|A′|. Applying Ruzsa span theorem(Theorem 1.2.3), we find that there exists a subspace V ⊂ Fn+m
2 which contains A′ suchthat |V | ≤ c|A′| where c = exp((1/δ)O(1)). We will use the subspace V to extract a linearfunction g : Fn2 → Fm2 such that dist(f, g) ≥ δO(1)/c. Setting parameters appropriately thetheorem will follow.
Define U = x ∈ Fn2 : (x, f(x)) ∈ V to be the projection of V to Fn2 . Note that Uis a subspace of Fn2 and that |U | ≥ |A′|. We can decompose the subspace V as the unionof subspaces Ug = (x, g(x)) : x ∈ U where g ∈ Linearn,m is a linear function. Since|V |/|U | ≤ c, we get that there exists a linear function g ∈ Linearn,m such that
|A′ ∩ Ug| ≥ (1/c) · |A′|.
15
This means that f is close to g since
|x ∈ Fn2 : f(x) = g(x)| ≥ (1/c) · |A′| ≥ (1/c) · δO(1) · 2n.
1.6 The polynomial Freiman-Ruzsa conjecture
The polynomial Freiman-Ruzsa conjecture is a conjecture on the structure of sets which havesmall doubling. Before stating the conjecture formally, it will be instructive to consider afew examples of sets of small doubling. We will focus for now on subsets of Fn2 , and later wewill comment on what changes in general Abelian groups.
Example 1.6.1. Let A ⊂ Fn2 be any set of size |A| = 2n/K. Then |A + A| ≤ K|A|. Thisexample shows us that A could be any large set, without specific structure.
Example 1.6.2. Let U, V ⊂ Fn2 be subspaces with U ∩ V = 0. Let u1, . . . , uK be linearlyindependent elements from U and define A = u1, . . . , uK+V . Then |A+A| ≤ O(K) · |A|.However, note that the span of A has size ≈ 2K |A|.
The second example shows that we cannot hope that small doubling means a polynomiallyrelated small span size. However, in this example, a large subset of A does have small spansize. This motivated the polynomial Freiman-Ruzsa conjecture. In the following, Span(A)denotes the linear subspace spanned by a set A.
Conjecture 1.6.3 (Polynomial Freiman-Ruzsa conjecture, over F2). Let A ⊂ Fn2 be a setwith |A + A| ≤ K|A|. Then there exists a subset A′ ⊂ A of size |A′| ≥ K−c|A| such thatSpan(A′) ≤ |A|, where c > 0 is an absolute constant.
The Polynomial Freiman-Ruzsa conjecture is one of the central conjectures in additivecombinatorics, since it allows for relations between two different notations of structure (smalldoubling, and small span of a large subset) with only a polynomial loss of parameters betweenthe two. It turns out that this conjecture is very useful for computer science application.Moreover, it has a large number of equivalent conjectures. For example, it show that inTheorem 1.5.3 the dependence of δ on ε can be improved to be polynomial instead of expo-nential. In fact, this is equivalent to the polynomial Freiman-Ruzsa conjecture. We give afew equivalent formulations below, for a more complete list see [24].
Lemma 1.6.4. The polynomial Freiman-Ruzsa conjecture (Conjecture 1.6.3) is equivalentto the following conjectures:
1. If A ⊂ Fn2 has |A + A| ≤ K|A| then there exists a subspace V ⊂ Fn2 of size |V | ≤ |A|such that |A ∩ V | ≥ K−c1 |A|.
2. If A ⊂ Fn2 has |A + A| ≤ K|A| then there exists a subspace V ⊂ Fn2 of size |V | ≤ |A|and a set X ⊂ Fn2 of size |X| ≤ Kc2 such that A ⊂ V +X.
16
3. If f : Fn2 → Fm2 has the property that Pr[f(x+ y) = f(x) + f(y)] ≥ 1/K then f is Kc3
close to a linear function (e.g, polynomial bounds in Theorem 1.5.3).
4. If f : Fn2 → Fm2 has the property that f(x + y) + f(x) + f(y) ∈ X for all x, y ∈ Fn2where |X| ≤ K, then f is Kc4 close to a linear function.
The best known results, getting close to Conjecture 1.6.3, were achieved by Sanders [51].We state below his result for Fn2 ; his result applies to general Abelian groups as well (withthe constant 4 replaces by 6). For interested readers, we refer to an exposition [34] of theproof over Fn2 .
Theorem 1.6.5 (Quasi-polynomial Freiman-Ruzsa theorem [51]). Let A ⊂ Fn2 be a set with|A + A| ≤ K|A|. Then there exists a subset A′ ⊂ A of size |A′| ≤ exp(c · log4K)|A| suchthat Span(A′) ≤ |A|, where c > 0 is an absolute constant.
We discuss briefly the extensions of Conjecture 1.6.3 to general Abelian groups. First,for groups of constant torsion (such as Fnp for constant p) the conjecture extends as is, exceptthat c is a function of p. However, for groups of large torsion (such as R) there is anotherexample which is worth noting.
Example 1.6.6. Let B ⊂ Rd be a large ball and let A = B ∩ Zd be the integer points in B.If B is large enough then |A| is roughly the volume of B. Similarly, |A+A| is approximatelythe volume of B+B, which is 2d times large than the volume of B. Hence, |A+A| ≈ 2d|A|.
The ball in the example can be replaced with any (nice enough) convex body. Thisexample tells us that in groups of large torsion there is another example of structured setswith low doubling constants: integer points in convex bodies. For general Abelian groups,the structure of Zd can be embedded in them via linear maps (that is, maps φ : Zd → G forwhich φ(x + y) = φ(x) + φ(y)). This is formulated by the following general version of thepolynomial Freiman-Ruzsa conjecture.
Conjecture 1.6.7 (Polynomial Freiman-Ruzsa conjecture, general Abelian groups). Let Gbe an Abelian group, A ⊂ G a set with |A+ A| ≤ K|A|. Then there exist
1. A subgroup H < G;
2. A convex body B ⊂ Rd for d ≤ c · logK, whose integer points are B ∩ Zd;
3. A linear map φ : Zd → G,
Such that for the set S = H + φ(B ∩ Zd) the following holds:
|S| ≤ |A|, |A ∩ S| ≥ K−c|A|.
where c > 0 is an absolute constant.
For example, in the special case of G = Fnp (where p could be large), Conjecture 1.6.7has the following immediate corollary.
17
Corollary 1.6.8. Let A ⊂ Fnp be a set with |A + A| ≤ K|A|. Then there exists a subset
A′ ⊂ A of size |A′| ≥ p−O(logK)|A| such that |Span(A′)| ≤ |A|.
Proof. Basically, the corollary holds since |φ(B ∩ Zd)| ≤ |φ(Zd)| ≤ pd. Formally, chooseA′ = A ∩ (H + b) for b ∈ φ(B ∩ Zd) which maximizes |A′|.
Another special case that is interesting is when G = Rn. Here, the subgroup H mustbe zero, and hence the entire structure comes from integer points in convex bodies. A weakversion of this conjecture is that a large subset of A has a low dimension. This conjecturewas explored in [16] with relations to some problems in lie groups.
Conjecture 1.6.9 (Polynomial Freiman-Ruzsa conjecture, weak version over Rn). Let A ⊂Rn be a set with |A + A| ≤ K|A|. Then there exist A′ ⊂ A of size |A′| ≥ K−c|A| such thatdim(A′) ≤ c · log(K), where c > 0 is an absolute constant.
1.7 Approximate duality
The polynomial Freiman-Ruzsa conjecture has found several applications in computer sci-ence. We focus here on a family of applications which utilize another useful concept, calledapproximate duality. It was introduced in computer science by Ben-Sasson and Ron-Zewi [41],who used it to give a construction of two-source extractors based on a specific constructionof Ramsey graphs. Later works used it to give an improvement on the log-rank conjecturein communication complexity [7] and to prove lower bounds on certain families of locallydecodable codes [8].
Let A,B ⊂ Fn be two subsets of a vector space. The sets A,B are called approximatedual if a noticeable fraction of the pairs a ∈ A, b ∈ B are orthogonal under the standardinner product on Fn. If the sets A,B were random and |F| = p we would expect 1/p fractionof the pairs to be orthogonal; hence, any larger fraction is considered noticeable.
Definition 1.7.1 (Approximate duality). Two sets A,B ⊂ Fn have duality measure ε if∣∣∣∣(a, b) ∈ A×B : 〈a, b〉 = 0∣∣∣∣ ≥ ( 1
|F|+ ε
)|A||B|.
We note that the definition also makes sense for F = R where 1/|F| = 0. The approximateduality conjecture is a conjecture on the structure of approximate dual sets. It speculatesthat any two large approximate dual sets must contain large subsets which are dual to eachother.
Conjecture 1.7.2 (Approximate duality conjecture [41]). Let A,B ⊂ Fn be sets with dualitymeasure ε. Then there exist subsets A′ ⊂ A,B′ ⊂ B such that
〈a, b〉 = 0 ∀a ∈ A′, b ∈ B′
and |A|/|A′|, |B|/|B′| ≤ exp(c(ε) ·√n), where c(ε) is a constant which depends on ε.
18
The bound |A|/|A′|, |B|/|B′| ≤ exp(c(ε) ·√n) might seem unnatural. However, it is the
best possible as can be seen by the following example.
Example 1.7.3. Fix any field F. Let A = B be the set of all vectors in 0, 1n of hammingweight (1/2)
√n. If we choose a ∈ A, b ∈ B uniformly, their expected inner product is
1/4. Hence, the probability that 〈a, b〉 = 0 is at least 3/4, e.g. A,B have duality measureε = (3/4) − 1/|F| ≥ 1/4. On the other hand, the largest subsets A′ ⊂ A,B′ ⊂ B which aredual are given by
A′ = x ∈ A : x1 = · · · = xn/2 = 0, B′ = x ∈ B : xn/2+1 = · · · = xn = 0.
It can be verified that |A|/|A′| = |B|/|B′| ≥ exp(c√n) for some constant c > 0.
We demonstrate the applicability of the approximate duality conjecture by the applicationof [41] who used it to construct two-source extractors from certain constructions of Ramseygraphs (also called dispersers). We give definitions below. We will denote by H = (U, V,E)a bi-partite graph with vertex sets U, V and edge set E ⊂ U ×V . For subsets A ⊂ U,B ⊂ Vwe denote by E(A,B) the set of edges between A and B, e.g. E(A,B) = E ∩ (A×B).
Definition 1.7.4 (Ramsey graph). A bi-partite graph H = (U, V,E) is an r-Ramsey graphif it contains no (r, r) complete set and no (r, r) independent set. That is, for all subsetsA ⊂ U,B ⊂ V of size |A|, |B| ≥ r,
1 ≤ |E(A,B)| ≤ |A||B| − 1.
Definition 1.7.5 (Two-source extractors). A bi-partite graph H = (U, V,E) is an (r, ε)two-source extractor if all the r × r induced graphs are balanced. Formally, for all subsetsA ⊂ U,B ⊂ V of size |A|, |B| ≥ r,
(1/2− ε)|A||B| ≤ |E(A,B)| ≤ (1/2 + ε)|A||B|.
Both Ramsey graphs and two-source extractors have been extensively studied in the graphtheory and computational complexity communities. The main question is what are the bestparameters possible. A simple probabilistic argument shows that if we choose a randomgraph with |U | = |V | = N and where each edge is chosen independently with probability1/2, then with good probability the resulting graph is an r = O(logN) Ramsey graph, andin fact also an (r = O(logN), ε) two-source extractor for any constant ε > 0. A majorline of research is trying to match these existential results by explicit constructions. Thebest quantitative results differ between explicit constructions of Ramsey graphs and of two-source extractors. Before describing the best results currently known, it will be instructiveto consider the following simple example first.
Example 1.7.6. Let N = 2n. Identify U = V = Fn2 and let E = (u, v) ∈ Fn2 × Fn2 :〈u, v〉 = 0. The resulting graph is a r = N1/2 + 1 Ramsey graph, since for any two subsetsA,B ⊂ Fn2 of size |A|, |B| > 2n/2 it cannot be that A,B are orthogonal to each other, and itcannot be that all the inner products are one. Hence, the induced graph on A×B is neither
19
complete nor empty. A slightly more involved argument using Fourier analysis shows thatthe same graph is also an (r, ε) two-source extractor for any ε > 0 and r = c(ε) ·N1/2 for anappropriate constant c(ε).
Thus, a non-trivial construction is one that beats the r = O(N1/2) bound. For Ramseygraphs, the best known construction is by Barak et al [3] who construct a Ramsey graphwith r = Nα for any α > 0 (in fact, they give r = N o(1)). For two source extractor, the bestknown construction is by Bourgain [12] who construct a two-source extractor for r = N1/2−α
for some tiny constant α > 0. Thus, it makes sense to see whether constructions of Ramseygraphs can somehow be converted to constructions of two-source extractors. The followingresult of [41] shows that this is possible, if the Ramsey graphs are constructed as sub-graphsof the simple example given above (in fact, most constructions have this property or a similarone) and if one assumes the approximate duality conjecture.
Theorem 1.7.7 ( [41]). Consider the graph H = (U, V,E) with vertex sets U, V ⊂ Fn2 andedge set E = (u, v) ∈ Fn2 × Fn2 : 〈u, v〉 = 0. Assume that the graph H is a r-Ramseygraph. Then, assuming the approximate duality conjecture (Conjecture 1.7.2), it is also a(s, ε) two-source extractor for any ε > 0 and s = r · exp(c(ε)
√n).
Proof. Assume towards contradiction that H is not an (s, ε) two-source extractor. Then,there exist subsets A ⊂ U,B ⊂ V of size |A|, |B| ≥ s such that
|E(A,B)− |A||B|/2| ≥ ε|A||B|.
Consider first the case that |E(A,B)| ≥ (1/2+ε)|A||B|. Then, we can apply the approximateduality conjecture and deduce that there exists subsets A′ ⊂ A,B′ ⊂ B which are orthogonaland |A|/|A′|, |B|/|B′| ≤ exp(c(ε) ·
√n). This means that |E(A′, B′)| = 0 and by assumption
this can only happen if |A′| < r or |B′| < r. Hence, we have s < r · exp(c(ε)√n). Now, if
instead |E(A,B)| ≤ (1/2 − ε)|A||B| we can apply the same argument to the complementgraph H ′ = (U, V,E ′ = (u, v) : 〈u, v〉 = 1). Note that H ′ is also a r-Ramsey graph, and inH ′ we have |E ′(A,B)| ≥ (1/2 + ε)|A|B|.
We do not know if the approximate duality conjecture is equivalent to the polynomialFreiman-Ruzsa conjecture, or if one of them follows from the other. In [41] it was shownthat certain weak variants of the two conjectures are equivalent. The best result to date isby [7] who proved that a weak version of the approximate duality conjecture over F2 holdsassuming the polynomial Freiman-Ruzsa conjecture over F2. This was generalized to groupsof constant torsion in [8] who used it to prove lower bounds for certain families of locallydecodable codes. Below, we give the version over F2.
First, we state a very weak version of Conjecture 1.7.2 which is known to hold.
Theorem 1.7.8 ( [41]). Let A,B ⊂ Fn2 be sets with duality measure 1− ε. Then there existsubsets A′ ⊂ A,B′ ⊂ B such that
〈a, b〉 = c ∀a ∈ A′, b ∈ B′
for some c ∈ F2, where |A|/|A′|, |B|/|B′| ≤ exp(−δn) for δ = O(ε log(1/ε)).
20
Proof. By assumption we know that Pra∈A,b∈B[〈a, b〉 = 1] ≤ ε. Let
A0 = a ∈ A : Prb∈B
[〈a, b〉 = 1] ≤ 2ε.
By Markov inequality we have |A0| ≥ |A|/2. Let d = dim(A0) and fix a1, . . . , ad ∈ A0 linearlyindependent. For each b ∈ B define w(b) to be
w(b) = i ∈ [d] : 〈ai, b〉 = 1.
We have that Eb∈B[w(b)] =∑d
i=1 Pr[〈ai, b〉 = 1] ≤ 2εd. Hence, if we define
B0 = b ∈ B : w(b) ≤ 4εd
then |B0| ≥ |B|/2. Now, the main observation is that the number of distinct inner productpatterns (〈ai, b〉)i∈[d] for b ∈ B0 is bounded by
(d≤4εd
)≤ exp(δd) for δ = O(ε log(1/ε)). Thus,
we can find c1, . . . , cd ∈ F2 such that for
B′ = b ∈ B0 : 〈ai, b〉 = ci ∀i ∈ [d]
we have |B′| ≥ |B0| exp(−δd). Note that since a1, . . . , ad span A0, we have that for anya ∈ A0, the inner products 〈a, b〉 for all b ∈ B′ are all equal. Let
A′ = a ∈ A0 : 〈a, b〉 = c ∀b ∈ B
where c ∈ F2 is chosen to maximize |A′| ≥ |A0|. The theorem follows.
An improved result, which assumes the polynomial Freiman-Ruzsa conjecture, is thefollowing.
Theorem 1.7.9 ( [7]). Let A,B ⊂ Fn2 be sets with duality measure ε. Assuming the polyno-mial Freiman-Ruzsa conjecture over F2 (Conjecture 1.6.3), there exist subsets A′ ⊂ A,B′ ⊂B such that
〈a, b〉 = 0 ∀a ∈ A′, b ∈ B′
and |A|/|A′|, |B|/|B′| ≤ exp(c(ε) · n/ log(n)), where c(ε) is a constant which depends on ε.
21
22
Chapter 2
Arithmetic progressions
2.1 Arithmetic progressions in large sets
A k-term arithmetic progression is a sequence x, x+y, x+2y, . . . , x+(k−1)y. A basic questionin number theory is the existence of arithmetic progressions in certain sets of numbers. Oneof the early results in this field is the Van der Waerden theorem [58] who showed that if thenumbers 1, 2, . . . , n are colored with a few colors, then one of the color sets must contain anarithmetic progression of any length, as long as n is large enough.
Theorem 2.1.1 (Van der Waerden Theorem [58]). Let c, k ≥ 1 be constants, and let n ≥n0(c, k) be large enough. Then for any coloring of the integers 1, . . . , n with c colors, thereexist a k-term arithmetic progression which is monochromatic (e.g. all its elements areassigned the same color).
It turns out that in fact, any large set of integers must contain a long arithmetic progres-sion. This was first proved by Roth [42, 43] for length 3-term arithmetic progressions, andlater generalized by Szemeredi [54] to k-term arithmetic progressions for any fixed k ≥ 3.
Theorem 2.1.2 (Szemeredi Theorem [54]). Let ε > 0, k ≥ 1 be constants, and let n ≥n0(ε, k) be large enough. Then any subset A ⊂ 1, . . . , n of size |A| ≥ εn contains a k-termarithmetic progression.
Clearly, Theorem 2.1.1 follows immediately from Theorem 2.1.2 by setting ε = 1/c andchoosing A to be the largest color class. This completes the picture from a qualitativeviewpoint when ε is a constant, but leaves open to problem of sub-constant ε. Otherwiseput, for any fixed k and large n, what is the smallest size of a subset in 1, . . . , n whichguarantees the existence of a k-term arithmetic progression. A famous conjecture of Erdosand Turan is that any set of natural numbers A for which the reciprocals diverge containsan arbitrary long arithmetic progression.
Conjecture 2.1.3. Let A ⊂ N be a set such that∑
n∈A1n
= ∞. Then A contains anarithmetic progression of every length.
23
The answer to these question is not well understood in general, even for k = 3. Theoriginal argument of Roth showed in fact that any subset A ⊂ 1, . . . , n of size n/ log log(n)must contain a 3-term arithmetic progression. Further improvements due to several au-thors [11,13,27,49,50,55] improved this bound to n/ log1−o(1) n. For the general Szemeredi’stheorem, the original argument gave a bound of n/ log · · · log n, where the log function isiterated a constant number of times (the constant depends on k). Of particular interest innumber theory are prime numbers. If the quantitative bounds were improved to n/ log(n)then they will immediately imply that the primes contain arbitrarily long arithmetic pro-gressions. Even though this quantitative improvement is still unknown, Green and Tao [25]proved via more specific methods that the primes contain arbitrary long arithmetic progres-sions.
We give below a proof for the simplest case of k = 3 case (Roth theorem). We willpresent a simplified version due to Meshulam [36], who obtained similar results for groupsof odd order. We will focus here on the simplest case, that of subsets of Fn3 .
Theorem 2.1.4 ( [36]). Let A ⊂ Fn3 be of size |A| ≥ c · 3n/n for an absolute constant c > 0.Then A contains a 3-term arithmetic progression. That is, there exist x, y ∈ Fn3 with y 6= 0such that
x, x+ y, x+ 2y ⊂ A.
A subset A ⊂ Fn3 which has no 3-term arithmetic progressions is called a capset. Equiv-alently, it is a set not containing any line. Theorem 2.1.4 showed that the maximal size ofa capset is |A| ≤ (c/n) · 3n. More recently, Bateman and Katz [4] improved the bound to|A| ≤ (c/n1+ε) ·3n for some small unspecified ε > 0. We do not know how tight these boundsare. A simple example fir a capset is A = 0, 1n which has size |A| = 2n. There are betterexamples, but they all have size |A| ≤ (3− ε)n for some ε > 0. Hence, there are large gapsbetween the known lower and upper bounds on the density of capsets.
2.1.1 Proof of Meshulam’s theorem (Theorem 2.1.4)
The proof utilizes Fourier analysis over Fn3 . This is very similar to the Fourier analysis overFn2 we introduced in Section 1.5.1 with a few small changes. The characters of Fn3 are thefunctions χα : Fn3 → C defined as
χα(x) = ω〈α,x〉
where ω = exp(2π/3) is a primite cubic root of unity, α ∈ Fn3 and 〈α, x〉 =∑n
i=1 αixi is thestandard inner product in Fn3 . Given a function F : Fn3 → C it can be decomposed as
F (x) =∑α∈Fn
3
F (α) · χα(x)
where
F (α) = 3−n∑x∈Fn
3
F (x) · χα(x).
24
Note that the characters are orthogonal under the inner product 〈F,G〉 =∑
x F (x)G(x);that χα(x+ y) = χα(x)χα(y); χα+β(x) = χα(x)χβ(x); and that by Parseval’s identity,∑
α∈Fn3
|F (α)|2 = 3−n∑x∈Fn
3
|F (x)|2.
Proof of Theorem 2.1.4. Let c > 0 be a large enough constant to be determined later. Weprove by induction on n that any set A ⊂ Fn3 of size |A| = (c/n)3n must contain a 3-termarithmetic progression. We shorthand σ = c/n for the density of A and F = 1A for theindicator function for A. We have that for any x, y ∈ Fn3 , y 6= 0,
F (x)F (x+ y)F (x+ 2y) = 0.
We will expand F to its Fourier coefficiets. Note that F (0) = σ. We say that the set A isε-uniform if for any other Fourier coefficient α 6= 0,
|F (α)| ≤ ε.
We will show that if A is uniform for a small enough ε then it must contain a 3-termarithmetic progression. In fact, it contains many arithmetic progressions. The reason is thatuniform sets behave in many ways like a random set of the same density. Now, a random setof size |A| = σ3n will contain approximately σ3 · 32n many 3-term arithmetic progressions.A uniform set will contain the same order of magnitude of 3-term arithmetic progressions.
Claim 2.1.5. If A is ε-uniform with ε < σ2/2 then A contains at least (σ3/4) · 32n many3-term arithmetic progressions.
Proof. Consider the expression
F (x)F (x+ y)F (x+ 2y).
It is equal to one when (x, x + y, x + 2y) is a 3-term arithmetic progression or when x ∈A, y = 0; and is equal to zero otherwise. Hence, if A contains less than (σ3/4) · 32n many3-term arithmetic progressions then
Ex,y∈Fn3[F (x)F (x+ y)F (x+ 2y)] ≤ σ3/4 + 3−n ≤ σ3/3.
On the other hand, the Fourier expansion gives
F (x)F (x+ y)F (x+ 2y) =∑
α,β,γ∈Fn3
F (α)F (β)F (γ)ω〈α,x〉+〈β,x+y〉+〈γ,x+2y〉.
Averaging over uniform x, y ∈ Fn3 , we get that the only contributions come when α+β+γ = 0and β+2γ = 0, e.g. when α = β = −2γ. Since we are over Fn3 , this is the same as α = β = γ.Hence
Ex,y∈Fn3[F (x)F (x+ y)F (x+ 2y)] =
∑α∈Fn
3
F (α)3.
25
We thus obtained that ∑α∈Fn
3
|F (α)|3 ≤ σ3/3.
We next separate the sum for α = 0 and α 6= 0. For α = 0 we have F (0) = σ. We boundthe sum over α 6= 0 using Parseval identity and the assumption that A is ε-uniform,∑
α∈Fn3 ,α 6=0
|F (α)|3 ≤ ε ·∑α∈Fn
3
|F (α)|2 = ε · Ex∈Fn3|F (x)|2 = ε · Pr[x ∈ A] = εσ.
We thus deduce thatσ3 − εσ ≤ σ3/3,
which implies ε ≥ (2/3) · σ2, which contradicts our assumption.
Hence, since we assume that A contains no 3-term arithmetic progressions, we knowthat there exists a Fourier coefficient α such that |F (α)| ≥ σ2/2. For v = 0, 1, 2 defineHv = x ∈ Fn3 : 〈α, x〉 = v to be co-dimension one affine subspaces orthogonal to α. Wewill next show that the density of A inside one of the affine subspaces Hv is somewhat largerthan the global density of A. This will then allow us to use induction applied to A ∩Hv.
Claim 2.1.6. Let A be a set of size |A| = σ3n. Assume there exist α 6= 0 such that
|F (α)| ≥ ε. Then there exists v ∈ 0, 1, 2 such that
|A ∩Hv| ≥ (σ + ε/6) · 3n−1.
Proof. Let σv = |A∩Hv|/3n−1 be the density of A∩Hv in Hv and let ∆v = σv−σ. We needto prove that ∆v ≥ ε/6 for some v ∈ 0, 1, 2. Clearly, ∆0 + ∆1 + ∆2 = 0. We have that
F (α) = σ0 + ωσ1 + ω2σ2 = ∆0 + ω∆1 + ω2∆2
where we used the fact that 1 + ω+ ω2 = 0. Hence, since |F (α)| ≥ ε such must exist u suchthat |∆u| ≥ ε/3. If ∆u ≥ ε/3 we are done; otherwise, since ∆0 + ∆1 + ∆2 = 0 there mustexist v ∈ 0, 1, 2 \ u such that ∆v ≥ ε/6.
We now conclude the proof by induction. Let c > 0 be a large enough constant and letA ⊂ Fn3 be a set without 3-term arithmetic progressions of size |A| = (c/n)3n. We provedthat there exist a co-dimension one affine subspace Hv such that
|A ∩Hv| ≥(c
n+
c2
12n2
)3n−1.
The set A∩Hv is also a set without 3-term arithmetic progressions in n− 1 dimensions. Bychoosing c large enough (c > 24 suffices) we have that
c
n+
c2
12n2≥ c
n− 1.
Hence, we found a set in n− 1 dimensions of size c/(n− 1) · 3n−1 without 3-term arithmeticprogressions, which violates the induction hypothesis.
26
2.2 Sets without arithmetic progressions
We next focus on constructions of large sets without arithmetic progressions. As we men-tioned, for subsets of Fn3 the best constructions have exponentially small density. If we turnour attention to subsets of the integers, much better constructions are known. We presentbelow a construction due to Behrend [5] from 1946, which has only been marginally improvedsince.
Theorem 2.2.1. There exist a set A ⊂ 1, . . . , n which contains no 3-term arithmeticprogressions of size |A| ≥ n · 2−O(
√logn).
Theorem 2.2.1 was generalized by Rankin [39] who gave a construction of a set A ⊂1, . . . , n which contains no k-term arithmetic progression of size |A| = n · 2−O((logn)1/(k−1)).
Before moving to the proof, we note that in some application it is easier to assume weare given a subset A ⊂ Zn without any 3-term arithmetic progressions. That is, we allow thearithmetic progression to be taken modulo n. A simple solution is to choose A ⊂ 1, . . . , n/3based on Theorem 2.2.1, e.g. without any 3-term arithmetic progressions over the integers.It is simple to verify that such a set cannot have 3-term arithmetic progressions modulo n.
Proof of Theorem 2.2.1. We start with the following basic observation: a sphere in Rd doesnot contain any 3 points on a line. In order to discretize this, define
B =x ∈ 0, . . . ,m− 1d :
∑x2i = r
.
Choosing a value of 0 ≤ r ≤ (m− 1)2d which maximizes |B| gives
|B| ≥ md
m2d.
Since B is a subset of the sphere of radius r1/2, it implies that B does not contain any 3-termarithmetic progression (and in fact no three points on a line). We next construct a map fromZd to Z which will preserve the property of having no 3-term arithmetic progression. Suchmaps are called Freiman homomorphisms. Define a map ϕ : [m]d → Z by
ϕ(x) =d∑i=1
xi · (2m)i−1.
We will take A = ϕ(x) : x ∈ B and we will show that A has no 3-term arithmeticprogressions. Assume to the contrary that there exist x, y, z ∈ [m]d such that ϕ(x), ϕ(y), ϕ(z)form a 3-term arithmetic progression. We will show that this implies that x, y, z also form a3-term arithmetic progression, which is impossible by the construction of B.
Now, if ϕ(x), ϕ(y), ϕ(z) form a 3-term arithmetic progression then ϕ(x) + ϕ(z) = 2ϕ(y),that is
S =d∑i=1
(xi + zi − 2yi) · (2m)i−1 = 0.
27
By construction we have that |xi + zi − 2yi| < 2m. Hence, since S mod 2m = 0 we musthave that x1 +z1−2y1 = 0. Since (S/2m) mod 2m = 0 we must have that x2 +z2−2y2 = 0,and so on. That is, x+ z = 2y, i.e. x, y, z form an arithmetic progression.
Finally, we optimize parameters. Set
n = max(A) ≤d−1∑i=0
(m− 1) · (2m)i−1 ≤ (m− 1) · (2m)d − 1
2m− 1≤ (2m)d.
Furthermore we have
|A| ≥ n
2dm2d.
Setting 2m = 2√
logn and d =√
log n so that (2m)d = n, we get that
|A| ≥ n · 2−3√
logn(1+o(1)).
2.3 Graphs with many disjoint triangles
We give a simple and beautiful application of the existence of large sets without arithmeticprogressions. We use them to construct graphs on n vertices with n2−o(1) edge disjointtriangles, and no other triangles. Clearly, the number of edge disjoint triangles is boundedby the number of edges n2, hence up to the o(1) term this is the best possible. Formally, weprove the following theorem.
Theorem 2.3.1 (Graph with many edge disjoint triangles). For every n, there exist a graphon n vertices with n2 · 2−O(
√logn) edge disjoint triangles, and no other triangles.
Let p be prime and let A ⊂ Zp be a set without any 3-term arithmetic progressionsmodulo p. We will construct a graph G on n = 3p vertices with p|A| edge disjoint triangles,and no other triangles (for other values of n add isolated vertices to the graph). ChoosingA as guaranteed by Theorem 2.2.1, the theorem follows.
Now, define G = (V,E) to be the desired graph with |V | = 3p. We partition the vertexset to three sets of size p each, and label the vertices xi, yi, zi for i ∈ Zp. We set the edgesto be
E = (xi, yi+a) : i ∈ Zp, a ∈ A ∪ (yi, zi+a) : i ∈ Zp, a ∈ A ∪ (xi, zi+2a) : i ∈ Zp, a ∈ A.
Clearly, the graph G = (V,E) contains p|A| triangles (xi, yi+a, zi+2a) for i ∈ Zp, a ∈ A. Byconstruction, these triangles are edge disjoint. What we need to show is that this accountsfor all the triangles in the graph G.
Claim 2.3.2. If (xi, yj, zk) is a triangle in G then j = i+ a, k = i+ 2a for some a ∈ A.
28
Proof. We have j = i+ a, k = j + b, k = i+ 2c for a, b, c ∈ A. Substituting gives
i+ a+ b = i+ 2c,
which implies that a, c, b form an arithmetic progression. Since A contains no non-trivialarithmetic progressions this can only hold if a = b = c.
The following theorem generalizes Theorem 2.3.1 by replacing triangles with cliques ofa larger size due to Dell and van Melkebeek [18]. Later, we will see an application of it incomplexity theory.
Theorem 2.3.3 (Graph with many edge disjoint s-cliques). Let s, t ≥ s2 be large enoughparameters. There exists a graph G on O(s · t1/2+o(1)) vertices such that
1. G contains t edge disjoint cliques of size s.
2. G contains no other cliques of size s.
Moreover, the graph can be computed by a deterministic poly-time algorithm.
We note that the parameters are tight, up to the o(1) factor: each clique of size s containsO(s2) edges, and hence t edge disjoint cliques of size s contain ts2 edges. Hence, the graphmust have at least s
√t vertices.
Proof. Let p ≈ t1/2+o(1) be prime. Let A ⊂ Zp be the set without 3-term arithmetic progres-sions guaranteed by Theorem 2.2.1, and assume that |A| = p1−o(1) ≥
√t. Define the graph
G = (V,E) as follows: V = [s]× Zp. For any a ∈ A, b ∈ Zp define the set of vertices
Ka,b = (i, ai+ b) : i ∈ [s],and let E be the union of cliques on Ka,b for all a ∈ A, b ∈ Zp. We observe a few propertiesof G:
(i) The graph G has |V | = s · p = O(s · t1/2+o(1)) vertices.
(ii) The graph G is s-partite with parts Vi = (i, c) : c ∈ Zp for 1 ≤ i ≤ s.
(iii) The graph G has p|A| ≥ t edge disjoint cliques Ka,b of size s each.
We need to show that any clique K of size s is one of Ka,b. Let K = (i, h(i)) : i ∈ [s] be aclique. We need to show that h(i) = ai+ b for some a ∈ A, b ∈ Zp.
Consider vertices in three consecutive parts v0 = (i, h(i)), v1 = (i + 1, h(i + 1)), v2 =(i+ 2, h(i+ 2)). By construction we have
• The vertices v0, v1 lie on the line (x, a1x+ b1). In particular, h(i+ 1)− h(i) = a1.
• The vertices v1, v2 lie on the line (x, a2x+ b2). In particular, h(i+ 2)− h(i+ 1) = a2.
• The vertices v0, v2 lie on the line (x, a3x+ b3). In particular, h(i+ 2)− h(i) = 2a3.
We thus get that a1 + a2 = 2a3, or equivalently that a1, a3, a2 is a 3-term arithmetic pro-gression. Since A has no nontrivial 3-term arithmetic progressions this could only hold ifa1 = a2 = a3. Hence also b1 = b2 = b3. Now, since this holds for any 3 consecutive points inthe clique, it must hold for all points in the clique.
29
2.4 Impossibility of compression of NP-hard languages
We present a complexity theoretic application of Theorem 2.3.3. We will present a resultof Dell and van Melkebeek [18] that under reasonable complexity assumptions, instances ofNP-hard problems cannot be non trivially compressed by poly-time algorithms. We focuson the clique problem, and note that similar results can be obtained for other NP-hardproblems.
The clique problem takes as input an undirected graph G and a parameter k, and askswhether G has a clique of size k.
CLIQUE = (G, k) : Graph G has a clique of size k.
The CLIQUE problem is one of the classic NP-hard problem [17, 29, 33] and the best al-gorithms for it run in exponential time. We study a different question: can the input beefficiently compressed while preserving the information of whether it is in CLIQUE or not.If the graph G has n vertices then the input can be specified using O(n2) bits. Clearly, if oneallows exponential time (or non deterministic polynomial time) then this can be done by sim-ply solving the CLIQUE problem. However, what we show is that deterministic polynomialtime algorithms cannot compress (G, k) to a witness of length n2−ε bits, for any ε > 0.
Theorem 2.4.1 ( [18]). Let ε > 0. Assume there exists a deterministic poly-time algorithmC which takes as input (G, k), where G is a graph on n vertices and 1 ≤ k ≤ n, and outputsw = C(G, k) such that
1. Given w it is possible to determine whether (G, k) ∈ CLIQUE or not (informationtheoretically, non efficiently).
2. The length of w is at most O(n2−ε).
Then coNP ⊂ NP/poly.
The conclusion of Theorem 2.4.1 is that statements of the form ”a CNF formula on nvariables has no satisfying assignment” have proofs of length poly(n). This seems unlikelygiven the current state of knowledge, as the shortest known proofs for such statements havelength exponential in n. We note (without proof) that Theorem 2.4.1 can be used to showthat other NP-hard problems cannot be compressed. For example, it can be used to showthat under the same assumptions, 3-CNF formulas cannot be compressed in deterministicpoly-time to witnesses of length O(n3−ε).
We prove Theorem 2.4.1 in the remainder of this section. We assume the existence ofa compression algorithm given in Theorem 2.4.1. Fix for the remainder of the proof anNP-complete language L, where we choose 3-SAT for convenience. Let s = O(n3) denotethe size of 3-CNF formulas (i.e. the number of clauses). For t = poly(s) large enough to bedetermined later, we define the language OR(t, s) to be the language given by disjunction oft formulas of size s each.
OR(t) = (ϕ1, . . . , ϕt) :ϕ1, . . . , ϕt are 3-CNF formulas with s clauses each
and at least one of ϕ1, . . . , ϕt is satisfiable.
30
We first follow a standard reduction from 3-SAT to the clique problem on s-partite graphs.In the following we consider graphs G = (V,E) which are s-partite graphs, with each partof size 3. That is, V = V1 ∪ . . . ∪ Vs, |V1| = . . . = |Vs| = 3 and E ⊂ ∪i 6=jVi × Vj.
Claim 2.4.2. Let ϕ be a 3-CNF formula with s clauses. Then there exist an s-partite graphGϕ, with each part of size 3, such that ϕ is satisfiable if and only if Gϕ has a clique of sizes. Moreover, Gϕ can be constructed in deterministic poly-time.
Proof. Let ϕ(x) = C1 ∧ . . . ∧ Cs where each Ci is a disjunction of three literals. Constructthe graph Gϕ = (V,E) where V = V1 ∪ . . . ∪ Vs, |V1| = . . . = |Vs| = 3, and the i-th vertex inVa is connected to the j-th vertex in Vb if the i-th literal in Ca and the j-th literal in Cb arenot contradicting. Then Gϕ has a clique of size s if and only if ϕ is satisfiable.
We will show that assuming the compression algorithm, for large enough t an instanceof OR(t) can be compressed to at most t bits. We then show that this type of compressionimplies that L ∈ coNP/poly, and hence NP ⊂ coNP/poly or equivalently coNP ⊂ NP/poly.
Lemma 2.4.3 (Compression of OR(t)). Let s be large enough and let t ≥ s6/ε. Assuming theconditions of Theorem 2.4.1 there exists a deterministic poly-time compression algorithm C ′,that given a collection of t 3-CNF formulas of size s produces a witness w = C ′(ϕ1, . . . , ϕt)such that
1. Given w it is possible to determine (non efficiently) whether at least one of ϕ1, . . . , ϕtis satisfiable.
2. The length of w is at most t.
Proof. Let G = (V,E) be a graph on n = O(s · t1/2+o(1)) vertices with t edge disjoint cliquesK1, . . . , Kt of size s, and no other cliques of size s, guaranteed by Theorem 2.3.3. Definea graph G′ = (V ′, E ′) as follows. Its vertex set is V ′ = V × 1, 2, 3. For each 1 ≤ i ≤ t,identify the vertices of Ki × 1, 2, 3 with the vertices of Gϕi
and set the edges accordingly(crucially, this uses the fact that K1, . . . , Kt are edge disjoint). Then, G′ contains a cliqueof size s if and only if at least one of ϕ1, . . . , ϕt is satisfiable.
The graph G′ can be built in deterministic poly-time. Hence, if we run the assumedcompression algorithm we get a witness w = C(G′) as required. The length of w is boundedby
n2−ε ≤ (s · t1/2+o(1))2−ε ≤ tε/3 · t1−ε/2+o(1) ≤ t1−ε/6+o(1) < t
for large enough values of t.
We finalize the proof by showing that compression of OR(t) implies that the basic lan-guage L defining it (in our case, 3-SAT instances of size s) is in coNP/poly.
Lemma 2.4.4. Let L ⊂ 0, 1∗ be a language, t = t(n) be polynomially bounded, and let
ORL(t) = (x1, . . . , xt) :x1, . . . , xt ∈ 0, 1n
and at least one of x1, . . . , xt is in L.
31
Assume that for large enough n, there exists a deterministic poly-time compression algorithmC such that w = C(x1, . . . , xt) allows to determine whether (x1, . . . , xt) ∈ ORL(t), and thelength of w is at most t(n). Then L ∈ coNP/poly.
Proof. Fix n and let U = Lc ∩ 0, 1n. Given x ∈ U we will construct a proof that x ∈ Uof size poly(n) which can be verified deterministically in poly-time. This will prove thatLc ∈ NP/poly, which is equivalent to L ∈ coNP/poly.
Let W = w : w = C(x1, . . . , xt), x1, . . . , xt ∈ U be the set of witnesses for inputs oflength n not in ORL(t). The basic idea is the following: if we are given as advice that somew ∈ W , and moreover it happen to be that C(x, x2, . . . , xt) = w for some x2, . . . , xt ∈ 0, 1n,then it must be the case that x ∈ U . This can be verified by a deterministic algorithm giventhe advice w.
Concretely, we will show that there exists a small list of advice strings w1, . . . , wn ∈ Wsuch that the following holds. For any x ∈ U , there exist x1, . . . , xt ∈ 0, 1n such that
1. C(x1, . . . , xt) = wi for some 1 ≤ i ≤ n.
2. x = xj for some 1 ≤ j ≤ t.
Then, given w1, . . . , wn as a common advice for all inputs on length n, the proof that x ∈ Uwill be the appropriate list (x1, . . . , xt). Clearly, this proof can be verified deterministicallygiven the common advice.
We construct the sequence w1, . . . , wn in a greedy manner. Let U1 = U . Since |W | ≤ 2t
there must exists a popular witness w1 that many x1, . . . , xt are compressed to. That is, theset
A1 = (x1, . . . , xt) ∈ (U1)t : C(x1, . . . , xt) = w1has size |A1| ≥ |U1|t/|W | = 2nt−t. An element x ∈ U is covered by w1 if there exist(x1, . . . , xt) ∈ U t such that x = xi for some 1 ≤ i ≤ t and C(x1, . . . , xt) = w1. Define the setof elements covered by w1 to be
B1 = x ∈ U1 : ∃(x1, . . . , xt) ∈ A1, x = xi for some 1 ≤ i ≤ t.Note that A1 ⊂ (B1)t. Hence |B1| ≥ |U1|/2. Now, if x happens to be in B1 then we aredone: we prove that x ∈ B1 by giving the appropriate sequence (x1, . . . , xt) as proof. LetU2 = U1 \B1 be the remaining potential inputs, with |U2| ≤ |U1|/2.
Following the same arguments with U1 replaced with U2, there exist a popular w2 ∈ Wsuch that
A2 = (x1, . . . , xt) ∈ (U2)t : C(x1, . . . , xt) = w2has size |A2| ≥ |U2|t/2t. The set of elements in U2 covered by w2 is
B2 = x ∈ U2 : ∃(x1, . . . , xt) ∈ A2, x = xi for some 1 ≤ i ≤ t.Again, A2 ⊂ (B2)t and hence |B2| ≥ |U2|/2. If x happens to be in B2 then we are done asbefore. Continuing in this fashion, we get a sequence w1, w2, . . . ∈ W where each one halvesthe potential inputs not already covered. As |U | ≤ 2n this process halts after at most nsteps.
Theorem 2.4.1 follows by combining Lemma 2.4.3 and Lemma 2.4.4.
32
Chapter 3
Sum-product phenomena
Let A be a set of numbers. Denote by A + A = a + a′ : a, a′ ∈ A its sumset and byA · A = aa′ : a, a′ ∈ A its productset. In general, if |A| = n then n ≤ |A + A|, |A ·A| ≤ n2. Consider the following two examples. If A is an arithmetic progression, sayA = 1, 2, 3, . . . , n, then |A+A| = O(n) and |A·A| = Ω(n2). If A is a geometric progression,say A = 1, 2, 4, 8, . . . , 2n, then |A ·A| = O(n) and |A+A| = Ω(n2). Is it possible that both|A + A|, |A · A| are small, of size O(n)? or is it true that whenever the sumset is small, theproductset is large, and vice versa? Results of this type are called ”sum product theorems”and have a multitude of applications in number theory, discrete geometry and complexity.We will present two results of this type in different domains: when A is a set of real numbers,and when A is a subset of a finite field. We refer the reader also to a mini course on additivecombinatorics [10] for exposition of the different applications of sum product theorems.
3.1 Sum-product theorems over the reals
We start by considering the case where A is a subset of real numbers. Erdos and Sze-meredi [20] shows that either the sumset or the productset of A is polynomially larger thanA.
Theorem 3.1.1 ( [20]). For any set A of real numbers, either |A+A| ≥ c|A|1+ε or |A ·A| ≥c|A|1+ε, where c, ε > 0 are absolute constants.
In fact, in the same paper they conjecture that one of |A+A|, |A ·A| should be close tomaximal.
Conjecture 3.1.2 ( [20]). For any ε > 0 and large enough set of real numbers A,
max(|A+ A|, |A · A|) ≥ |A|2−ε.
Despite much research, Conjecture 3.1.2 remains open. The best known result is bySolymosi [52] who showed that max(|A + A|, |A · A|) ≥ |A|4/3−o(1). Another related recent
33
result is by Iosevich et al [28] who showed that |A · A+ A · A| ≥ |A|2−o(1). This is tight (upto the o(1) factor), e.g. by taking A to be a geometric progression.
We prove Thoerem 3.1.1 in the remainder of this section. The proof will rely on theordering of the real numbers. We will later see more algebraic proofs in the context of finitefields, which do not rely on this. We first establish a lemma, which proves the result in thespecial case where all the numbers are bounded between m and 2m for some value m.
Lemma 3.1.3. Let A be a set of real numbers between m and 2m. Then
max(|A+ A|, |A · A|) ≥ c|A|1+ε,
where c, ε > 0 are absolute constants.
Proof of Lemma 3.1.3. Let a1 < . . . < aN be the elements of A. Partition A to N7/8
segments of length s = N1/8 each, Ai = ais, . . . , ais+s−1. Let B = Ai be the segment of thesmallest diameter (e.g. max(Ai)−min(Ai)). We claim that if i− j > 10 then Ai+B,Aj +Bare disjoint, and Ai · B,Aj · B are disjoint. To see that, let a ∈ Ai, a′ ∈ Aj, b, b′ ∈ B and leta = a′ + x, b = b′ + y. By assumption, x ≥ 9 · diam(B) ≥ 9|y|. Clearly, it cannot be thata+ b = a′+ b′. Assume that ab = a′b′. Then (a′+x)b = a′(b−y) which implies x/y = −a′/b.But |x/y| ≥ 9 and 1/2 ≤ a′/b ≤ 2 since the elements of A are between m and 2m.
Thus, we can bound
|A+ A|+ |A · A| ≥∑i
|A10i +B|+ |A10i ·B|.
Let I = i : |Ai + B|, |Ai · B| ≤ s1+α for α > 0 to be chosen later (α < 1/3 suffices). Wewill shortly show that |I| ≤ s4 = N1/2 which is much smaller than the number of sets A10i
which is (1/10) ·N7/8. Hence, for most sets Ai, |Ai + B| + |Ai · B| ≥ s1+α = N1/8·(1+α) andwe deduce that
|A+ A|+ |A · A| ≥ (N7/8/10) ·N1/8·(1+α) = (1/10) ·N1+α/8.
To conclude the proof, fix i ∈ I such that |Ai + B|, |Ai · B| ≤ s1+α. We will show thatthere exist b1, b2, b3, b4 ∈ B such that
(b1 − b2)b4
b3 − b4
∈ Ai.
As there are |B|4 = s4 choices for b1, b2, b3, b4, this shows that |I| ≤ s4.In order to show this, consider the s2 sums a+ b for a ∈ Ai, b ∈ B. By assumption, there
are at most s1+α distinct possible sums. Hence, there is a sum S such that
aj + bj = S
for at least s1−α different pairs aj ∈ Ai, bj ∈ B. Consider now the s2−2α products ajbk. Asthey take at most s1+α different values, if α < 1/3 then there is a pair which takes the sameproduct. This means we can find a1, a2 ∈ Ai, b1, b2, b3, b4 ∈ B such that
a1 + b1 = a2 + b2, a1b3 = a2b4.
This set of equations has at most one solution. Solving for a1 gives the claim.
34
Proof of Theorem 3.1.1 given Lemma 3.1.3. Let |A| = N . We can assume, without loss ofgenerality, that all the elements in A are positive (by possibly reducing |A| by a factor oftwo). Let Ai = a ∈ A : 2i < a ≤ 2i+1. Note that |Ai+Ai| and |Aj+Aj| are disjoint for anyi 6= j: any element in Ai + Ai is in (2i+1, 2i+2] and any element in Aj + Aj is in (2j+1, 2j+2].Similarly |Ai · Ai| and |Aj · Aj| are disjoint for i 6= j. Let I = i : 0 < |Ai| < N1/4. Weconsider two cases.
(i) Assume first that∑
i∈I |Ai| < N/2. Then by removing these elements (and reducing|A| by a factor of two) we can assume that if Ai is not empty then |Ai| ≥ N1/4.Applying Lemma 3.1.3, we get that for any non-empty set Ai,
|Ai + Ai|+ |Ai · Ai| ≥ c|Ai|1+ε ≥ c|Ai|N ε/4
and
|A+ A|+ |A · A| ≥∑i
|Ai + Ai|+ |Ai · Ai| ≥ cN ε/4∑|Ai| = cN1+ε/4.
(ii) Otherwise, we must have that |I| ≥ (1/2) · N3/4. Let I ′ ⊂ I be such that |i − j| ≥ 2for all i, j ∈ I ′, where |I ′| ≥ (1/4) · N3/4. Fix arbitrary ai ∈ Ai for i ∈ I ′. Then, thepairwise sums ai + aj for i, j ∈ I ′ are all disjoint. Otherwise, if ai + aj = ak + a` wherei > max(j, k, `) then ai > 2i while ak + a` ≤ 2i−1 + 2i−3 < 2i. Hence, we get that
|A+ A| ≥(|I ′|2
)≥ c ·N3/2.
3.2 Sum-product theorems over finite fields
We consider here now the sum product problem over finite fields. Let F be a finite field.There is another source of examples of finite fields, coming from sub-fields. Specifically, if Kis a sub-field of F then K + K = K · K = K. Hence, we will restrict our attention to fieldswithout nontrivial subfields, namely prime finite fields F = Fp. We note that results can beextended to non-prime fields if care is given to exclude sub-fields (or sets close to sub-fields).Bourgain, Kats and Tao [15], Konyagin [30] and Bourgain, Glibichuk and Konyagin [14]proved a sum-product theorem over prime finite fields. We will later see applications of thistheorem for the construction of extractors in complexity theory.
Theorem 3.2.1 ( [14,15,30]). For any α > 0 there exist ε > 0 such that the following holds.For any prime p, and any set A ⊂ Fp of size |A| ≤ p1−α,
max(|A+ A|, |A · A|) ≥ |A|1+ε.
35
For concreteness lets fix α = 0.1. The proof of Theorem 3.2.1 is based on two mainlemmas. First, we will show that there exist a polynomial expression (an expression ofthe form A · A + A − A) which grows polynomially in A. Then, we show that if both|A + A| ≤ |A|1+ε and |A · A| ≤ |A|1+ε then there is a large subset of A which doesn’t growunder rational expressions. Together these yield a contradiction.
Lemma 3.2.2. Let A ⊂ Fp of size |A| ≤ p0.9. There exists a polynomial expression R(A)such that |R(A)| ≥ |A|1+δ, where δ > 0 is a constant.
Lemma 3.2.3. Let A ⊂ Fp be such that |A + A|, |A · A| ≤ |A|1+ε. For any polynomialexpression R there exists a subset B ⊂ A, of size |B| ≥ |A|1−c(R)ε, such that
|R(B)| ≤ |B|1+c(R)ε.
Here, c(R) is a constant which depends just on the polynomial R.
We first describe how the proof of Theorem 3.2.1 follows from Lemma 3.2.2 andLemma 3.2.3, and the proceed to prove the two lemmas.
proof of Theorem 3.2.1. Let A ⊂ Fp be of size |A| ≤ p0.9 such that |A+A|, |A ·A| ≤ |A|1+ε.Let R be the polynomial expression in Lemma 3.2.2. By Lemma 3.2.3 there exists a subsetB ⊂ A of size |B| ≥ |A|1−cε, with c = c(R), such that |R(B)| ≤ |B|1+cε. But by Lemma 3.2.2,|R(B)| ≥ |B|1+δ. Hence ε ≥ δ/c.
3.2.1 Proof of Lemma 3.2.2
Consider the set of λ ∈ Fp for which |A + λA| < |A|2. That is, there exist a1, a2, a3, a4 ∈ Asuch that
a1 + λa2 = a3 + λa4,
or equivalently
λ =a3 − a1
a2 − a4
.
Let B = A−AA−A be this set. Assume first that B = Fp. To get a polynomial expression take
R0(A) = (A − A)(A − A). Then by Ruzsa triangle inequality applied to the multiplicativegroup F∗p, we have
|R0(A)| ≥(|B||A|
)1/2
|A| ≥ p1/20|A|.
Otherwise assume B 6= Fp. Fix a nonzero a ∈ A. There must exists λ ∈ B such thatλ+ a /∈ B, where crucially we use the fact that the additive group of Fp is generated by anynonzero element. Then
|A+ (λ+ a)A| = |A|2.Let λ = (a3 − a1)/(a2 − a4), then
|(a2 − a4)A+ (a3 − a1 + aa2 − aa4)A| = |A|2.
36
HenceR1(A) = (A− A)A+ (A− A+ A · A− A · A)A
have size |R1(A)| ≥ |A|2. Now, R(A) = R0(A) +R1(A) grows polynomially for any A.
3.2.2 Proof of Lemma 3.2.3
We will need the following lemma.
Lemma 3.2.4. Let A be a subset of an Abelian group such that |A+A| ≤ K|A|. Then thereexists B ⊂ A of size |B| ≥ K−O(1)|A| such that any b1 − b2 ∈ B −B can be written as
b1 − b2 =12∑i=1
ai, ai ∈ A ∪ −A
in at least K−O(1)|A|11 distinct ways; and any b1 + b2 ∈ B +B can be written as
b1 + b2 =12∑i=1
ai, ai ∈ A ∪ −A
in at least K−O(1)|A|11 distinct ways.
Proof. We will prove the lemma for B − B, the proof for B + B is similar. Let N = |A|.Following the same idea of the proof of the BSG Theorem (Theorem 1.4.1), let C ⊂ A + Abe the set of elements which can be written as c = a1 + a2 with a1, a2 ∈ A in at least N/2Kdistinct ways. As |A+A| ≤ K|A| we have |C| ≥ N/2K. Let H be the bi-partite graph withvertex sets A,A and edges E = (a1, a2) : a1 + a2 ∈ C. Applying Lemma 1.4.2 to H, letB,B′ ⊂ A, |B|, |B′| ≥ K−O(1)N be sets such that for any b ∈ B, b′ ∈ B′ there are at leastK−O(1)N2 paths of length three (b, a, a′, b′) between b, b′. This means that we can write
b+ b′ = (b+ a)− (a+ a′) + (a′ + b′) = c1 − c2 + c3
with c1, c2, c3 ∈ C in at least K−O(1)N2 distinct ways. As any element c ∈ C can be writtenas c = a+ a′ in at least N/2K distinct ways, we can write
b+ b′ =6∑i=1
ai, ai ∈ A ∪ −A
in at least K−O(1)N5 distinct ways. To conclude express any b1 − b2 ∈ B − B as (b1 + b′)−(b2 + b′) over all choices of b′, which gives that we can express
b1 − b2 =12∑i=1
ai, ai ∈ A ∪ −A
in at least K−O(1)N11 distinct ways.
37
In order to prove Lemma 3.2.3 we will show by induction that more and more complexpolynomials do not expand.
Lemma 3.2.5. Fix t, s ≥ 1. There exists B ⊂ A of size |B| ≥ |A|1−cε such that
|(Bt −Bt) ·BsB−s| ≤ |B|1+cε,
where c = c(t, s).
Lemma 3.2.3 follows from Lemma 3.2.5, since for s = 0 we have |Bt−Bt| ≤ |B|1+cε, andby the the Plunneke-Ruzsa theorem (Theorem 1.3.1) this implies that also |tBt − tBt| ≤|B|1+c′ε for c′ = 2tc. Hence, for any polynomial expression R(·), if we choose t large enoughwe can express R(B) as the sum or difference of at most t monomials of degree t, hence|R(B)| ≤ |B|1+c(R)·ε. We now prove Lemma 3.2.5.
Proof. We will prove the lemma by induction on t. The base case is t = 1. Let B ⊂ A bethe set guaranteed by Lemma 3.2.4. Any b1 − b2 ∈ B −B can be written as
b1 − b2 =12∑i=1
ai, ai ∈ A ∪ −A
in at least |A|11−O(ε) distinct ways. Multiplying by an arbitrary element of BsB−s, we getthat any element x ∈ (B −B)BsB−s can be written as
x =12∑i=1
xi, xi ∈ ±As+1A−s
in at least |A|11−O(ε) distinct ways. Since |A · A| ≤ |A|1+ε, by the Plunneke-Ruzsa theorem(Theorem 1.3.1) we have |As+1A−s| ≤ |A|1+O(εs). Hence the number of choices for x1, . . . , x12
is |A|11+O(εs), and hence
|(B −B) ·BsB−s| ≤ |A|11+O(εs) = |B|11+O(εs).
For the inductive case, assume we proved the lemma for t, ` for all ` ≥ 0; we will prove itfor t+ 1, s. Let ` = `(t, s) be large enough (to be determined later) and assume we alreadyconstructed a set B ⊂ A of size |B| ≥ |A|1−cε such that
|(Bt −Bt) ·B`B−`| ≤ |B|1+cε.
In particular, |B ·B| ≤ |B|1+cε. Apply Lemma 3.2.4 to the multiplicative group of Fp to geta subset C ⊂ B of size |C| ≥ |B|1−O(cε), such that any x ∈ C · C can be written as
x =12∏i=1
bi, bi ∈ B ∪B−1
38
in at least |B|11−O(cε) distinct ways. Furthermore, lets assume (without loss of generality)that in all these b1 ∈ B. Multiplying this expression by an arbitrary element of Ct−1, we canwrite any element in x ∈ Ct+1 as
x =12∏i=1
bi, b1 ∈ Bt, b2, . . . , b12 ∈ B ∪B−1
in at least |B|11−O(cε) distinct ways.Fix now x, y ∈ Ct+1 and a representation x = x1 . . . x12. We will enumerate over rep-
resentations y = y1 . . . y12, with x1, y1 ∈ Bt and xi, yi ∈ B ∪ B−1 for i = 2, . . . , 12. Wehave
x− y = x1x2 . . . x12 − y1y2 . . . y12
= (x1 − y1)x2x3 . . . x12
+ y1(x2 − y2)x3 . . . x12
+ . . .
+ y1y2 . . . y11(x12 − y12)
All the terms are contained in (Bt − Bt)Bt+11B−11. If we multiply the expression by anarbitrary element of BsB−s, then any term would be contained in (Bt − Bt)B`B−` for` = t+ s+ 11. In particular, any element of (Ct+1−Ct+1)CsC−s is expressable in |B|11−O(cε)
ways as the sum of 12 terms, each coming from a domain of size |B|1+cε. Hence
|(Ct+1 − Ct+1) · CsC−s| ≤ |C|1+c′ε
for c′ = O(c).
3.3 Extractors
Extractors are deterministic functions which can ”extract” pure randomness from ”weak”random sources. There are various notions of extractors, and here we will focus on extractorsfor independent sources. First, we need some notion of wntropy which measures the amountof randomness in a random variable or a distribution. There are several potential notions.Let X be a random variable. We define
• The Erdos-Renyi entropy of X is log(1/Pr[X1 = X2]) where X1, X2 are i.i.d copies ofX. Equivalently, it is log(1/
∑x Pr[X = x]2).
• The Shannon entropy of X is∑
x Pr[X = x] log(1/Pr[X = x]).
• The min-entropy of X is minx log(1/Pr[X = x]).
For example, if X is uniform on a set of size N , then it has entropy logN is all the abovenotions of entropy. Hence, we think of a random variable with entropy k as a source whichhides k bits of randomness. Here, we focus on min-entropy for concreteness, although theresults can be generalized to other notions of entropy. The following claim is useful.
39
Claim 3.3.1. Let X be a random variable with min-entropy k. Then X is the convexcombination of uniform distributions on sets of support size 2k. That is, there exists setsA1, A2, . . . each of size 2k, such that if Xi ∈ Ai is chosen uniformly then
Pr[X = x] =∑
pi Pr[Xi = x]
where∑pi = 1.
A two-source extractor is a function which takes as input two independent inputs, eachwith some nontrivial amount of min-entropy, and outputs a bit which is close to uniform (oreven more than one bit, but here we will stick with the simplest notions).
Definition 3.3.2 (Two-source extractor). A function E : 0, 1n × 0, 1n → 0, 1 is a(k, ε) extractor if, for any two independent random variables X, Y , taking values in 0, 1n,each with min-entropy at least k,
1/2− ε ≤ Pr[E(X, Y ) = 0] ≤ 1/2 + ε.
If k = δn we say that E has min-entropy rate δ.
Given Claim 3.3.1, in order to verify that E(·, ·) is a two-source extractor, it will beenough to verify it whenever X, Y are independent and each uniformly distributed on a setof size 2k. As we already noted before, a two-source extractor can equivalently be viewedas a bi-partite graph H = (U, V,E) with U = V = 0, 1n and E = (x, y) : E(x, y) = 0.However, here we would prefer the functional definition given above.
We note that if one chooses the function E uniformly, then a simple probabilistic ar-gument shows that it is an extractor for k = O(log n). Our goal will try to come close tothis with explicit constructions. We will see that there is a huge gap between the resultsthat we can achieve explicitly, and the parameters that a random construction gives. Inparticular, we will see that Hadamard two-source extractor which would be an extractorfor k = O(n1/2), and a construction of Bourgain [12] which improves this to k = O(n1/2−η)for some small η > 0. Along the way, we will see that achieving extractors for multipleindependent sources is an easier problem, and for which we will describe an extractor dueto Barak et al [2] which achieves k = δn with a constant number of sources (the constantdepends on δ).
3.3.1 The Hadamard two-source extractor
We start by describing one of the most basic two-source extractor, the Hadamard (or innerproduct) extractor. It is defined as
EHad(x, y) = 〈x, y〉
where the inner product is computed modulo 2. We will show that the Hadamard extractorperforms well for min-entropy rates above 1/2. However, it breaks down at min-entropy 1/2.
40
For example, one can take X to be uniform on a subspace of dimension n/2, and Y to beuniform on the orthogonal subspace.
We next prove that the Hadamard extractor extracts random bits from min-entropy ratemore than 1/2. To recall, based on Claim 3.3.1, it suffices to prove this for X, Y the uniformdistributions over sets A,B, respectively, where |A|, |B| 2n/2. In fact, we will show thatthis extractors works whenever |A||B| 2n.
Lemma 3.3.3. Fix ε > 0. Let A,B ⊂ Fn2 be sets such that |A||B| ≥ (1/ε)3 · 2n. Ifx ∈ A, Y ∈ B are chosen independently and uniformly, then
1/2− ε ≤ Prx∈A,y∈B
[EHad(x, y) = 0] ≤ 1/2 + ε.
Proof. We will prove the equivalent statement∣∣Ex∈A,y∈B[(−1)〈x,y〉]∣∣ ≤ 2ε.
Define
S =α ∈ Fn2 :
∣∣Ey∈B[(−1)〈α,y〉]∣∣ ≥ ε
.
We will show that S cannot be too big, in particular |S| ≤ ε|A|, and hence
∣∣Ex∈A,y∈B[(−1)〈x,y〉]∣∣ ≤ 1
|A|
|S|+ ∑x∈A\S
∣∣Ey∈B[(−1)〈x,y〉]∣∣ ≤ 2ε.
In order to bound S, define the function ϕB(x) = 2n
|B|1B(x), and note that its Fourier coeffi-cients are exactly
ϕB(α) = Ey∈Fn2[ϕB(y)(−1)〈α,y〉] = Ey∈B[(−1)〈α,y〉].
Hence,
S = α ∈ Fn2 : |ϕB(α)| ≥ ε.
By Parseval identity, we know that∑α∈Fn
2
|ϕB(α)|2 = Ey∈B[ϕB(y)2] =2n
|B|.
In particular, as |ϕB(α)| ≥ ε for all α ∈ S, we must have that
|S| ≤ (1/ε2) · 2n
|B|≤ ε|A|.
41
3.3.2 Extractors for several independent sources
Breaking the min-entropy rate barrier of 1/2 achieved by the Hadamard extractor turns outto be a hard problem. As a first step, we will show that if we are allows to have more than 2independent sources, then the sum-product theorem will allow us to construct extractor fromany constant min-entropy rate. This is a result of Barak, Impagliazzo and Wigderson [2].First, we formally define multi-source extractors, which are an immediate generalization oftwo-source extractors.
Definition 3.3.4 (Multi-source extractors). A function E : (0, 1n)` → 0, 1 is an `-source (k, ε) extractor, if for any ` independent random variables X1, . . . , X` supported on0, 1n, each with min entropy at least k,
1/2− ε ≤ Pr[E(X1, . . . , X`) = 0] ≤ 1/2 + ε.
Let us recall the sum-product theorem we proved. Let A be a subset of a prime finitefield Fp of size |A| ≤ p0.9. We proved in Theorem 3.2.1 that max(|A + A|, |A · A|) ≥ |A|1+ε
for some ε > 0. Hence
|A · A+ A · A| ≥ |A|1+ε.
We can now apply this again, and deduce that either |A · A+ A · A| ≥ p0.9, or else
|(A · A+ A · A) · (A · A+ A · A) + (A · A+ A · A) · (A · A+ A · A)| ≥ |A|(1+ε)3 .
Applying this again and again, if we started with |A| = pδ, we will get to a set of size≥ p0.9 after constantly many steps (the constant depends on δ). Hence, we get the followingcorollary.
Corollary 3.3.5. For any δ > 0 there exist ` = `(δ) such that the following holds. Thereexists a polynomial expression R : F`p → Fp, such that for any subset A ⊂ Fp of size |A| ≥ pδ,
|R(A,A, . . . , A)| ≥ p0.9
Barak et al [2] prove a statistical analog of Corollary 3.3.5. Analogously to the 0, 1n,we say that a random variable X taking values in Fp has min-entropy rate δ if minx Pr[X =x] ≤ p−δ.
Theorem 3.3.6 ( [2]). For any δ > 0 there exist ` = `(δ) such that the following holds.There exists a polynomial expression R : F`p → Fp, such that for any independent randomvariables X1, . . . , X` ∈ Fp, each with min-entropy rate at least δ,
R(X1, . . . , X`)
is statistically close to a distribution with min-entropy rate 0.9.
42
In order to conclude and extract a random bit, we apply the Hadamard extractor on twoindependent copies R(X1, . . . , X`), R(X`+1, . . . , X2`). We defined the Hadamard extractorover Fn2 . In order to apply it, let n be such that 2n−1 < p < 2n, and fix an arbitrary injectivemap Ψ : Fp → 0, 1n. A subset S ⊂ Fp of size p0.9 is mapped to a subset Ψ(S) ⊂ 0, 1nof size approximately 20.9n. This puts us in a position to apply the Hadamard two-sourceextractor to two independent copies of an image of R.
To conclude, the multi-source extractor takes as input independent random variablesX1, . . . , X2` ∈ Fp, each with min-entropy rate at least δ, and outputs
〈Ψ(R(X1, . . . , X`)),Ψ(R(X`+1, . . . , X2`))〉.
3.4 Two source extractor for min-entropy rate below
1/2
The Hadamard two-source extractor stops working at min-entropy rate 1/2. The reason isthat one can take X uniform over a subspace of Fn2 of dimension n/2, and Y uniform overthe orthogonal subspace. Then both X, Y have min-entropy rate 1/2 but 〈X, Y 〉 = 0 withprobability one. However, this is a pathological example, and typically we do not expect suchhighly structured sources. If one can encode the source to a source which is ”unstructured”,then maybe one can achieve a two-source extractor for a lower min-entropy rate. This is theidea behind the two-source extractor of Bourgain [12], which we discuss in this section. Hisconstruction achieves a min-entropy rate of 1/2−ε0 for some small ε0 > 0. It is a challengingopen problem to find explicit constructions which achieve better rates. We also refer thereader to an exposition of this result by Rao [40].
As a first step, we show that if the sources grow with addition, then one can achieve abetter min-entropy rate. We recall that it is sufficient to consider random variables whichare uniform over sets. For a set A, define the collision probability of its k-iterated sum as
CPk(A) = Pra1,...,ak,a
′1,...,a
′k∈A
[a1 + . . .+ ak = a′1 + . . .+ a′k],
We have CP1(A) = |A|−1 and CPk+1(A) < CPk(A) for all k. The following lemma showsthat, if for some fixed k, CPk(A)CPk(B) = O(2−n), then the Hadamard extractor workssuccessfully for the sets A,B.
Lemma 3.4.1. Fix k ≥ 1. Let A,B ⊂ Fn2 be sets such that CPk(A)CPk(B) ≤ εc(k)2−n,where c(k) is a constant depending only on k. Then∣∣Ea∈A,b∈B(−1)〈a,b〉
∣∣ ≤ ε.
Proof. We will prove the lemma for k = 2, the proof of the general case is similar. First, wenote that by Cauchy-Schwarz,(
Ea∈A,b∈B[(−1)〈a,b〉])2 ≤ Ea∈A
(Eb∈B[(−1)〈a,b〉]
)2= Ea∈A,b,b′∈B[(−1)〈a,b+b
′〉].
43
Applying Cauchy-Schwarz again gives that(Ea∈A,b∈B[(−1)〈a,b〉]
)4 ≤ Ea,a′∈A,b,b′∈B[(−1)〈a+a′,b+b′〉].
Hence, it suffices to bound Ea,a′∈A,b,b′∈B[(−1)〈a+a′,b+b′〉]. Note that CP2(A) is the collisionprobability for the random variable a + a′, and similarly CP2(B) for b + b′. We will seethat these can replace the role that |A|, |B| took in the analysis of the Hadamard extractor.Define
S =α ∈ Fn2 :
∣∣∣Eb,b′∈B[(−1)〈α,b+b′〉]∣∣∣ ≥ ε
.
Denote by p = Pra,a′∈A[a+ a′ ∈ S], where we will show that p ≤ ε. As before, this will showthat
Ea,a′∈A,b,b′∈B[(−1)〈a+a′,b+b′〉] ≤ Pr[a+ a′ ∈ S] +∑x/∈S
Pr[a+ a′ = x]∣∣∣Eb,b′∈B[(−1)〈x,b+b
′〉]∣∣∣ ≤ 2ε.
Define ϕ(x) = 2n · Prb,b′∈B[b + b′ = x]. Then ϕ(α) = Eb,b′∈B[(−1)〈α,b+b′〉] and hence
S = α : |ϕ(α)| ≥ ε. We can bound
|S|ε2 ≤∑α∈Fn
2
|ϕ(α)|2 = Ex∈Fn2[ϕ(x)2] = 2n
∑x∈Fn
2
Prb,b′∈B
[b+ b′ = x]2 = 2nCP2(B).
Hence by assumption |S| ≤ εO(1)CP2(A)−1. On the other hand, we can bound
Pra,a′∈A
[a+ a′ ∈ S]2 = (∑x∈S
Pra,a′∈A
[a+ a′ = x])2 ≤ |S|∑x∈S
Pra,a′∈A
[a+ a′ = x]2 ≤ |S|CP2(A).
We conclude as this implies that
Pra,a′∈A
[a+ a′ ∈ S] ≤ (|S|CP2(A))1/2 ≤ εO(1).
So, we get that the only sets A,B for which the Hadamard extractor fails are those thatdo not grow under addition. The next idea is to build an encoding function which preventsthis situation from occurring. Assume we had a function f : Fn2 → Fm2 with the followingproperty: for any set A ⊂ Fn2 of size |A| ≥ 2δn, we have that
CPk(f(A)) ≤ εc(k)2−m/2
for some k ≥ 1. Then, if we define
E(X, Y ) = 〈f(X), f(Y )〉
we would get a two-source extractor for min-entropy rate δ.
44
Note that equivalently, if we view this extractor as a bi-partite graph, then it is a sub-graph of the graph corresponding to the Hadamard extractor over Fm2 . That is, it is abi-partite graph of the form (U, V,E) where |U | = |V | = 2n, U, V ⊂ 0, 1m, and E =(u, v) : 〈u, v〉 = 0.
The only remaining question is how to construct such an encoding function f . This iswhere the sum-product theorem will come into play. First, for technical reasons we will needto generalize the construction of the Hadamard extractor to fields of odd characteristics.The following Lemma is a generalization of Lemma 3.4.1. For a proof see [40].
Lemma 3.4.2. Let Fp be a prime finite field. Define E : Fnp × Fnp → 0, 1 as
E(x, y) = 〈x, y〉 mod 2.
If X, Y are independent random variables, taking values in Fnp , such that CPk(A)CPk(B) ≤εc(k)p−n, then ∣∣Pr[E(X, Y ) = 0]− 1/2
∣∣ ≤ ε+O(1/p).
In particular, if we choose p ≥ 1/ε we get similar guarantees to that of Lemma 3.4.1.Now, let p 6= 2, and consider the map f : Fp → F2
p defined as
f(x) = (x, x2).
Let A ⊂ Fp be mapped to f(A) ⊂ F2p. Consider first f(A) + f(A). For any x, y ∈ Fp, the
number of solutions for (a+ b, a2 + b2) = (x, y) is at most two, hence |f(A) +f(A)| ≥ |A|2/2.For the same reason, CP2(A) ≤ 2|A|−2. It might seem like we gained in the collisionprobability, but the universe size has also grown from p to p2, hence the min-entropy rateof A ⊂ Fp is the same as what is gained from CP2(f(A)) in F2
p. The actual advantage willcome once we consider the collision probability of the sum of three copies of f(A).
Lemma 3.4.3. There exist ε0 > 0 such that the following holds. If A ⊂ Fp has size |A| ≥p1/2−ε0 then CP3(f(A)) ≤ p−(1+ε0).
We are now in business. Combining Lemma 3.4.3 with Lemma 3.4.2, we get the followingcorollary.
Corollary 3.4.4. Fix prime p ≈ 1/ε. Define EBou : Fp × Fp → 0, 1 as
EBou(x, y) = (xy + x2y2) mod 2.
Then, if X, Y ⊂ Fp are random variables of min-entropy rate ≥ 1/2− ε0, then
|Pr[EBou(X, Y ) = 0]− 1/2| ≤ p−O(ε0).
We are left with proving Lemma 3.4.3. Surprisingly perhaps, its proof relies on theintersection pattern of lines and points in F2
p. Let P ⊂ F2p be a set of points in F2
p. A line in
45
F2p is set of the form `a,b = (x, ax+ b) : x ∈ Fp. Let L denote the set of all lines, |L| = p2.
Let L ⊂ L be a subset of the lines. We denote the set of intersections of P and L by
I(P,L) = (x, `) ∈ P × L : x ∈ L.
A trivial upper bound is |I(P,L)| ≤ |P ||L| ≤ p4. A slightly less trivial upper bound, utilizingthe fact that any two points lie on at most one line, is that |I(P,L)| ≤ |P |1/2|L| ≤ p3.
Claim 3.4.5. I(P,L) ≤ |P |1/2|L|.
Proof. We have I(P,L) =∑
x∈P,`∈L 1x∈`. Hence by Cauchy-Schwarz,
I(P,L)2 ≤
( ∑x∈P,`∈L
1x∈`
)2
≤ |L|∑`∈L
(∑x∈P
1x∈`
)2
= |L|∑x,x′∈P
∑`∈L
1x,x′∈` ≤ |L||P |2.
A similar claim, using the fact that two lines intersect in at most one point, gives thedual bound |I(P,L)| ≤ |P ||L|1/2. These bounds are in general tight, as if one takes P = F2
p
the set of all points and L = L the set of all lines, then |P | = |L| = p2 and |I(P,L)| = p3.However, what the next theorem shows is that the bound is tight only in such extreme cases.
Theorem 3.4.6 ( [14,15,30]). For any α > 0 there exist ε > 0 such that the following holds.If max(|P |, |L|) = M ≤ p2−α then |I(P,L)| ≤M3/2−ε.
The sum-product Theorem 3.2.1 follows from Theorem 3.4.6. To see this, let A ⊂ Fp bea set of size |A| ≤ p0.9. For simplicity, we assume 0 /∈ A. Define the following sets of pointsand lines
P = (A · A)× (A+ A), L = `a−1,b : a ∈ A, b ∈ A.
Any line `a−1,b ∈ L can be equivalently written as
`a−1,b = (x, a−1x+ b) : x ∈ Fp = (ax, x+ b) : x ∈ Fp.
Hence, whenever x ∈ A we have (ax, x+ b) ∈ P and hence
|I(P,L)| ≥ |L||A| = |A|3.
Let M = max(|P |, |L|) = |A · A||A + A|. If M ≥ p1.9 then we are done. Otherwise, applyTheorem 3.4.6 with α = 0.1. There exist ε > 0 for which |I(P,L)| ≤M3/2−ε, or equivalently
|A+ A| |A · A| ≥ (|A|3)1/(3/2−ε) = |A|2+Ω(ε).
One can also deduce Theorem 3.4.6 from Theorem 3.2.1, but this is more complicated, andwe refer the interested readers to the original papers for the details.
Finally, we show how Lemma 3.4.3 follows from Theorem 3.4.6.
46
Proof of Lemma 3.4.3. An element of f(A) + f(A) + f(A) is of the form
(a+ b+ c, a2 + b2 + c2)
with a, b, c ∈ A. For x, y ∈ Fp let
N(x, y) = (a, b, c) ∈ A3 : (a+ b+ c, a2 + b2 + c2) = (x, y).
We have
CP3(f(A)) =∑x,y∈Fp
Pra,b,c∈A
[(a+ b+ c, a2 + b2 + c2) = (x, y)]2 = |A|−6∑x,y∈Fp
N(x, y)2.
Let us first gather some intuition. Our goal is to show that CP3(f(A)) p−1. A trivialupper bound is N(x, y) ≤ 2|A|, since for every fixed c, there are at most two solutions for(a + b + c, a2 + b2 + c2) = (x, y). Moreover, since
∑N(x, y) = |A|3 this implies the bound
CP3(f(A)) ≤ O(|A|−2), which would require |A| p1/2, corresponding to min-entropy rate1/2. However, we are interested in beating this, e.g. in sets of size |A| ≥ p1/2−ε for someε > 0. This should not be surprising, as so far we did not apply any sum-product of point-lineincidence machinery.
Formally, for δ > 0 to be defined later, define the set of points to be
P = (x, y) ∈ F2p : N(x, y) ≥ |A|1−δ.
Since∑N(x, y) = |A|3 we have |P | ≤ A2+δ. If (a + b + c, a2 + b2 + c2) = (x, y) then
(a+ b+ c, ab+ ac+ bc) = (x, (y − x2)/2). We define the set of lines to be
L =`a+b,ab−(a+b)2 : a, b ∈ A
=(a+ b+ x, ab+ (a+ b)x) : x ∈ Fp : a, b ∈ A
.
We have |L| = |A|2. By our assumption, any point in P lies on at least |A|1−δ lines, hence
|I(P,L)| ≥ |P | · |A|1−δ.
We next will apply Theorem 3.4.6. Let M = max(|P |, |L|). Note that as we may assume|A| ≤ p1/2 we have M ≤ p1+2δ, which is in the allowed regime for any δ < 1/2. Moreconcretely, if we choose δ < 1/4 then M ≤ p3/2, and there exists an absolute constant ε > 0(independent of δ !) such that
|I(P,L)| ≤M3/2−ε.
Rearranging,|P | ≤ |A|(2+δ)(3/2−ε)−(1−δ) = |A|2−2ε+O(δ).
Choosing δ > 0 small enough, we have
|P | ≤ |A|2−ε.
47
We next derived an improved bound on CP3(f(A)).
|A|6 · CP3(f(A)) =∑
(x,y)∈P
N(x, y)2 +∑
(x,y)/∈P
N(x, y)2
≤ |P | · max(x,y)∈P
N(x, y)2 +∑
(x,y)/∈P
N(x, y) · max(x,y)/∈P
N(x, y)
≤ |P | · (2|A|)2 + |A|3 · |A|1−δ
≤ 4|A|4−ε + |A|4−δ.
By possibly replacing δ with min(ε, δ), we get that
CP3(f(A)) ≤ O(|A|−(2+δ)).
One can now verify that the lemma follows with ε0 = δ/6.
48
Bibliography
[1] A. Balog and E. Szemeredi. A statistical theorem of set addition. Combinatorica,14(3):263–268, 1994.
[2] B. Barak, R. Impagliazzo, and A. Wigderson. Extracting randomness using few inde-pendent sources. SIAM Journal on Computing, 36(4):1095–1118, 2006.
[3] B. Barak, A. Rao, R. Shaltiel, and A. Wigderson. 2-source dispersers for sub-polynomialentropy and ramsey graphs beating the frankl-wilson construction. In Proceedings of thethirty-eighth annual ACM symposium on Theory of computing, pages 671–680. ACM,2006.
[4] M. Bateman and N. Katz. New bounds on cap sets. Journal of the American Mathe-matical Society, 25(2):585–613, 2012.
[5] F. A. Behrend. On sets of integers which contain no three terms in arithmetical progres-sion. Proceedings of the National Academy of Sciences of the United States of America,32(12):331, 1946.
[6] M. Bellare, D. Coppersmith, J. Hastad, M. Kiwi, and M. Sudan. Linearity testing overcharacteristic two. 42(6):1781–1795, November 1996.
[7] E. Ben-Sasson, S. Lovett, and N. Ron-Zewi. An additive combinatorics approach relatingrank to communication complexity. In Foundations of Computer Science (FOCS), 2012IEEE 53rd Annual Symposium on, pages 177–186. IEEE, 2012.
[8] A. Bhowmick, Z. Dvir, and S. Lovett. New bounds for matching vector families. InProceedings of the 45th annual ACM symposium on Symposium on theory of computing,pages 823–832. ACM, 2013.
[9] M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to nu-merical problems. J. Comput. Syst. Sci., 47:549–595, 1993. Earlier version in STOC’90.
[10] L. T. Boaz Barak and A. Wigderson. A mini-course on additive com-binatorics. http://www.cs.princeton.edu/theory/index.php/Main/
AdditiveCombinatoricsMinicourse, 2007.
49
[11] J. Bourgain. On triples in arithmetic progression. Geometric and Functional Analysis,9(5):968–984, 1999.
[12] J. Bourgain. More on the sum-product phenomenon in prime fields and its applictaions.International Journal of Number Theory, 01(01):1–32, 2005.
[13] J. Bourgain. Roths theorem on progressions revisited. Journal d’Analyse Mathematique,104(1):155–192, 2008.
[14] J. Bourgain, A. Glibichuk, and S. V. Konyagin. Estimates for the number of sumsand products and for exponential sums in fields of prime order. Journal of the LondonMathematical Society, 73(2):380–398, 2006.
[15] J. Bourgain, N. Katz, and T. Tao. A sum-product estimate in finite fields, and appli-cations. Geometric & Functional Analysis GAFA, 14(1):27–57, 2004.
[16] M.-C. Chang. Some consequences of the polynomial freiman–ruzsa conjecture. ComptesRendus Mathematique, 347(11):583–588, 2009.
[17] S. A. Cook. The complexity of theorem-proving procedures. In Proceedings of the thirdannual ACM symposium on Theory of computing, pages 151–158. ACM, 1971.
[18] H. Dell and D. van Melkebeek. Satisfiability allows no nontrivial sparsification unlessthe polynomial-time hierarchy collapses. In Proceedings of the 42nd ACM symposiumon Theory of computing, pages 251–260. ACM, 2010.
[19] J. M. Deshouillers, F. Hennecart, and A. Plagne. On small sumsets in (Z/2Z)n. Com-binatorica, 24(1):53–68, 2004.
[20] P. Erdos and E. Szemeredi. On sums and products of integers. Studies in pure mathe-matics, pages 213–218, 1983.
[21] C. Even-Zohar. On sums of generating sets in Zn2 . Combinatorics, Probability andComputing, 21:916–941, 2012.
[22] G. Freiman. Foundations of a structural theory of set addition, volume 37. AmericanMathematical Society, Translations of Mathematical Monographs, 1973.
[23] W. Gowers. A new proof of Szemeredi’s theorem. Geometric And Functional Analysis,11(3):465–588, 2001.
[24] B. Green. The polynomial Freiman-Ruzsa conjecture, 2005.http://www.dpmms.cam.ac.uk/˜bjg23/papers/PFR.pdf.
[25] B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions. toappear. Ann. Math, 2005.
50
[26] B. Green and T. Tao. Freiman’s theorem in finite fields via extremal set theory. Comb.Probab. Comput., 18:335–355, May 2009.
[27] D. R. Heath-Brown. Integer sets containing no arithmetic progressions. Journal of theLondon Mathematical Society, 2(3):385–394, 1987.
[28] A. Iosevich, O. Roche-Newton, and M. Rudnev. On an application of guth-katz theorem.arXiv preprint arXiv:1103.1354, 2011.
[29] R. M. Karp. Reducibility among combinatorial problems. Springer, 1972.
[30] S. Konyagin. A sum-product estimate in fields of prime order. arXiv preprintmath/0304217, 2003.
[31] S. V. Konyagin. On the Freiman theorem in finite fields. Mathematical Notes, 84:435–438, 2008.
[32] I. Laba. Fuglede’s conjecture for a union of two intervals. Proceedings of the AmericanMathematical Society, 129(10):2965–2972, 2001.
[33] L. A. Levin. Universal sequential search problems. Problemy Peredachi Informatsii,9(3):115–116, 1973.
[34] S. Lovett. An exposition of sanders quasi-polynomial freiman-ruzsa theorem. ElectronicColloquium on Computational Complexity (ECCC), 19, 2012.
[35] S. Lovett and C. Even-Zohar. The freiman-ruzsa theorem in finite fields. 2012.
[36] R. Meshulam. On subsets of finite abelian groups with no 3-term arithmetic progres-sions. Journal of Combinatorial Theory, Series A, 71(1):168–172, 1995.
[37] G. Petridis. New proofs of plnnecke-type estimates for product sets in groups. Combi-natorica, 32(6):721–733, 2012.
[38] H. Plunneke. Eigenschaften und abschatzungen von wirkingsfunktionen. Gesellschaftfur Mathematik und Datenverarbeitung, 1969.
[39] R. A. Rankin. Sets of integers containing not more than a given number of terms inarithmetical progression. Proceedings of the Royal Society of Edinburgh. Section A.Mathematical and Physical Sciences, 65(04):332–344, 1961.
[40] A. Rao. An exposition of bourgains 2-source extractor. In Electronic Colloquium onComputational Complexity (ECCC), volume 14, 2007.
[41] N. Ron-Zewi and E. Ben-Sasson. From affine to two-source extractors via approximateduality. In Proceedings of the 43rd annual ACM symposium on Theory of computing,pages 177–186. ACM, 2011.
51
[42] K. Roth. Sur quelques ensembles dentiers. CR Acad. Sci. Paris, 234:388–390, 1952.
[43] K. F. Roth. On certain sets of integers. Journal of the London Mathematical Society,1(1):104–109, 1953.
[44] R. Rubinfeld and M. Sudan. Robust characterizations of polynomials and their appli-cations to program testing. Technical report, Ithaca, NY, USA, 1993.
[45] I. Z. Ruzsa. Arithmetical progressions and the number of sums. Periodica MathematicaHungarica, 25(1):105–111, 1992.
[46] I. Z. Ruzsa. An analog of Freiman’s theorem in groups. Structure Theory of Set-Addition. Asterisque, 258:323–326, 1999.
[47] A. Samorodnitsky. Low-degree tests at large distances. In Proceedings of the thirty-ninthannual ACM symposium on Theory of computing, STOC ’07, pages 506–515, New York,NY, USA, 2007. ACM.
[48] T. Sanders. A note on freiman’s theorem in vector spaces. Comb. Probab. Comput.,17:297–305, March 2008.
[49] T. Sanders. On roth’s theorem on progressions. arXiv preprint arXiv:1011.0104, 2010.
[50] T. Sanders. On certain other sets of integers. Journal d’Analyse Mathematique,116(1):53–82, 2012.
[51] T. Sanders. On the Bogolyubov-Ruzsa lemma. Analysis & PDE, 5(3):627–655, 2012.
[52] J. Solymosi. Bounding multiplicative energy by the sumset. Advances in mathematics,222(2):402–408, 2009.
[53] B. Sudakov, E. Szemeredi, and V. H. Vu. On a question of erdos and moser. DukeMathematical Journal, 129(1):129–155, 2005.
[54] E. Szemeredi. On sets of integers containing no k elements in arithmetic progression.Acta Arith., 27:199–245, 1975.
[55] E. Szemeredi. Integer sets containing no arithmetic progressions. Acta MathematicaHungarica, 56(1):155–158, 1990.
[56] T. Tao and V. H. Vu. Additive combinatorics, volume 13. Cambridge University Press,2006.
[57] E. Viola. Selected results in additive combinatorics: An exposition. Theory of Comput-ing, Graduate Surveys, 2:1–15, 2011.
[58] B. Waerden. van der.(1927). beweis einer baudetschen vermutung. Niew Arch, 1927.
52