chapter 2. a lattice point is a point in r d with integer coordinates. later we will talk about...
TRANSCRIPT
Lattices and Minkowski’s Theorem
Chapter 2
Geometry of Numbers
Number Theory
Preface
A lattice point is a point in Rd with integer coordinates.
Later we will talk about general lattice point.
Lattice Point
Let C ⊆ Rd be symmetric around the origin, convex, bounded and suppose that volume(C)>2d. Then C contains at least one lattice point different from 0.
Minkowski’s Theorem
Definitions* A C set is convex whenever x,y∊C implies
segment xy∊C .* An object C is centrally around the origin if
whenever (0,0) ∊ C and if x∊C then -x∊C.
Examples (d=2)Vol=2*2=4<22=4Vol=4*4=16>22=4
Proof
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Claim
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C’
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Proof –Claim(1)
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:Q Define
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C’
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2M
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Proof –Claim(2)
d
d
M
DCvol
CvolMCvolQ
)12
121()'(
)'()12()'(2D)(2Mvol(K)
:Hence C'. ofdiameter thedenotes D where
D]MD,-[-MK cube enlarged in the contained all are
They well.asdisjoint are s translate theseof every two thusand
,assumption the toaccording C' fromdisjoint is latesuch transEach
d
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Volume(cube) Possibilites of v in [-
M,M]d
K2M+2D
Upperbound
Proof –Claim(3)
ion.contradictA
M. oft independenamount certain aby
1 exceedingnumber fixed a is 1)(2)vol(C'hand,other On the
M. largely sufficientfor
1 toclose arbitrarly is side hand-right on the expression The
1)12
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d-
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Proof-Minkowski’s Theorem
theorem.sMinkowski' proves which C, that vmeans
'.2
1)(
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1
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1 have weso and too,C'in lies x)-x(v
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v).(C'C'point x a choose uslet claim, the toAccording
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ExampleLet K be a circle of diameter 26 meters centered at the origin. Trees of diameter 0.16 grow at each lattice point within K except for the origin, which is where Shrek is standing. Prove Shrek can’t see outside this mini forest.
Proof
theorem.sMinkowski'
scontradict which 4,16.40.16*26(C)But volume
origin. but the points lattice no contains
convex) itself is setsconvex ofon intersecti (The SKCset convex symetric theMeaning,
sight). thebolcked have would treea (otherwise
origin for theexcept K in point lattice no contains line middle theas
l with 0.16 width of S strip that themeans This origin. ethrough th
passing l line some along outside see couldShrek than Suppose
K
D=26m
D=0.16m
S
l
PropositionApproximating an irrational number by a fraction
nN
1
n
m- and Nn
such that n m, numbers natural ofpair a exists Then there
number. natural a N andnumber real a be (0,1)Let
Note: This proposition implies that there are infinitely many pairs m,n such that:
2
1||
nm
m
Proof
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n
m- Meaning,
.m)y n,(x 1
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theorem,sMinkowski' toaccording thereforeand 4N
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General Lattices
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TheoremMinkowski’s theorem for general lattices
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Proof(1)
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of nscombinatiolinear integer as written becan Fin lying of points
all then ,z,...,z,zby spanned subspace ldimensiona 1)-(i thedenotes F
If d.constructebeen already haveproperty following with thez,...,z,z
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Proof(2)
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Proof(3)
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Question…
How efficiently can one actually compute a nonzero lattice point in a symmetric convex body?
hard.-NP be known to is problem theinput, theofpart a as considered is d If
time.polynomialin solved becan problemsuch constant, a as considered is d If
An application in Number Theory
Theorem
.,
,ap :squares twoof sum a as written becan 4) (mod 1p primeEach 22
Zba
b
LemmaIf p is a prime with p ≡ 1(mod 4) then -1 is a quadric residue modulo p.
For a given positive integer n, two integers a and b are called congruent modulo n, written a ≡ b (mod n) if a-b is divisible by n.For example, 37≡57(mod 10) since 37-57=-20 is a multiple of 10.
Definitions-Number Theory
.nonresidue quadratic a is a Otherwise,
p). (mod with x*Fan x exists
thereif p modulo residue quadratic a called is *Faelement An
{0}.\FF*let and number, prime a is p where
p, modulo classes residue of field for the stand GF(p)FLet
2 a
Example: 42≡6(mod 10) so 6 is a quadratic residue (mod 10).
Proof(Theorem)
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).(mod0)1(2)(a calculate We
).,z of definition(by jpiqb and ia that meanswhich
Z,ji, somefor izb)(a, time,same At the .2a0 have We
{0}.\b)(a,point a contains C so and ,4det 4pp2 is C of area The
2p}.yx:Ry){(x, Cdisk the
for lattices generalfor theoremsMinkowski' use Wep.det have We
p).(0,z and q)(1, z where),,(z lattice heConsider t
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0≣q2≣-1(mod p)