one hundred years of uniform distribution theory hermann...

One Hundred Years of Uniform Distribution TheoryHermann Weyl’s Seminal Paper of 1916

Christoph [email protected]

TU Graz, Austria

MCQMC 2016, Stanford University

August 18, 2016

Table of contents

1 Hermann Weyl

2 Prehistory of uniform distribution theory

3 Weyl’s paper of 1916

4 Highlights and open problems

Christoph Aistleitner One Hundred Years of Uniform Distribution Theory

Section 1

Hermann Weyl

Hermann Weyl (1885–1955)

Hermann Weyl

David Hilbert and Hermann Weyl

Hermann Weyl

Bust of Hermann Weyl at the ETH Zurich

Hermann Weyl

Weyl riding a seesaw

Born 1885 in Elmshorn, Germany.

Places of work:

1904–1913: Gottingen (student of Hilbert)

1913–1930: ETH Zurich

1930–1933: Gottingen (successor of Hilbert)

1933–1951: IAS Princeton

Died 1955 in Zurich.

Topics:

Distribution of eigenvalues

Geometric foundations of manifolds and physics

Topological groups, Lie groups and representation theory

Harmonic analysis and analytic number theory

Foundations of mathematics

Weyl fermions

Work II

Topics named after Hermann Weyl (Wikipedia)

The Greatest Mathematicians of All Time

Citations

Section 2

Prehistory of uniform distribution theory

Prehistory of uniform distribution theory I

Theorem (Kronecker’s approximation theory, 1884)

Let α be an irrational number. Then the sequence ({nα})n≥1 isdense in the unit interval.

Prehistory of uniform distribution theory II

Definition

A sequence of real numbers x1, x2, . . . is called uniformlydistributed modulo one (u.d. mod 1) if

limN→∞

N∑n=1

1[0,a]({xn}) = a

for all a ∈ [0, 1].

Prehistory of uniform distribution theory III

Theorem (Bohl; Sierpinski; Weyl; 1909)

The sequence (nα)n≥1 is u.d. mod 1 if and only if α 6∈ Q.

Bohl: existence of mean motion in orbital mechanics

Sierpinski: approximation of real numbers by Farey fractions

Weyl: heat compensation in a circular ring with two components

Connection was realized by Bernstein in 1912.

Prehistory of uniform distribution theory III

Theorem (Bohl; Sierpinski; Weyl; 1909)

The sequence (nα)n≥1 is u.d. mod 1 if and only if α 6∈ Q.

Bohl: existence of mean motion in orbital mechanics

Sierpinski: approximation of real numbers by Farey fractions

Weyl: heat compensation in a circular ring with two components

Connection was realized by Bernstein in 1912.

Section 3

Weyl’s paper of 1916

Theorem (Functional analysis definition)

A sequence (xn)n≥1 is u.d. mod 1 if and only if for every 1-periodicRiemann integrable function f

limN→∞

N∑n=1

f (xn) =

0f (x) dx .

holds.

⇒ Hidden idea 1: Uniformly distributed sequences may be used for the

numerical approximation of integrals!

⇒ Hidden idea 2: Together with the uniform distribution of (nα)n≥1, precursor

of Birkhoff’s ergodic theorem.

limN→∞

N∑n=1

f (xn) =

0f (x) dx .

holds.

limN→∞

N∑n=1

f (xn) =

0f (x) dx .

holds.

limN→∞

N∑n=1

f (xn) =

0f (x) dx .

holds.

⇒ Hidden idea 3: Topological definition – assume that f is continuous.

“Die einfachste Funktion von der Periode 1 ist e2πix .”

(The simplest function with period 1 is e2πix .)

Theorem (Weyl’s criterion)

A sequence (xn)n≥1 is u.d. mod 1 if and only if

limN→∞

N∑n=1

e2πihxn = 0

holds for all integers h 6= 0.

⇒ Establishes connection between uniform distribution theory and exponential

Hardy–Littlewood: “The proof depends on a simple but ingenious use of the

theory of approximation to arbitrary functions by finite trigonometric

polynomials.”

limN→∞

N∑n=1

e2πihxn = 0

polynomials.”

limN→∞

N∑n=1

e2πihxn = 0

polynomials.”

limN→∞

N∑n=1

e2πihxn = 0

polynomials.”

Proof that (nα)n≥1 is u.d. mod 1 for irrational α:

We have

N∑n=1

e2πihnα =1

Ne2πihα

1− e2πihNα

1− e2πihα→ 0,

since hα 6∈ Z.

Proof that (nα)n≥1 is u.d. mod 1 for irrational α:

We have

N∑n=1

e2πihnα =1

Ne2πihα

1− e2πihNα

1− e2πihα→ 0,

since hα 6∈ Z.

“We remark that whenever the limit equation

limN→∞

N∑n=1

1[0,a]({xn}) = a

holds for all a ∈ [0, 1], then it holds uniformly for all a.”

⇒ Hidden idea: Take supremum over all intervals. Starting point of

quantitative theory?

“We remark that whenever the limit equation

limN→∞

N∑n=1

1[0,a]({xn}) = a

holds for all a ∈ [0, 1], then it holds uniformly for all a.”

⇒ Hidden idea: Take supremum over all intervals. Starting point of

quantitative theory?

Multidimensional theory

Continuous uniform distribution

Interpretation in terms of probability theory

Theorem (Uniform distribution of polynomial sequences)

Letxn = αkn

k + αk−1nk−1 + · · ·+ α0.

Then the sequence (xn)n≥1 is u.d. mod 1 if and only if at least oneof the coefficients αk , αk−1, . . . , α1 is irrational.

⇒ Hidden idea: Fundamental theorem of the theory of uniform distribution:

(xn)n≥1 is u.d. if all difference sequences (xn+h − xn)n≥1 are u.d. (van der

Corput, 1931).

Theorem (Uniform distribution of polynomial sequences)

Letxn = αkn

k + αk−1nk−1 + · · ·+ α0.

Then the sequence (xn)n≥1 is u.d. mod 1 if and only if at least oneof the coefficients αk , αk−1, . . . , α1 is irrational.

⇒ Hidden idea: Fundamental theorem of the theory of uniform distribution:

(xn)n≥1 is u.d. if all difference sequences (xn+h − xn)n≥1 are u.d. (van der

Corput, 1931).

Multidimensional polynomial sequences

Characterization of multidimensional uniform distribution

Connections with geometry of numbers

Sequences with multi-indices

Theorem (Metric theorem for parametric sequences)

Let a1, a2, . . . be a sequence of distinct positive integers. Then thesequence

(anx)n≥1

is u.d. mod 1 for almost all x .

“ Even if I believe that the value of such theorems, where an undetermined

exceptional set of measure zero appears, is not high, I will still give a proof of

this assertion.”

⇒ Metric number theory.

(anx)n≥1

this assertion.”

(anx)n≥1

this assertion.”

Appendix on compact manifolds

⇒ Hidden idea: Uniform distribution in compact spaces and uniform

distribution in topological groups.

Appendix on compact manifolds

⇒ Hidden idea: Uniform distribution in compact spaces and uniform

distribution in topological groups.

Section 4

Highlights and open problems

Highlights and open problems I

Definition

Let x1, . . . , xN be points in [0, 1]. Then the number

D∗N(x1, . . . , xN) = supa∈[0,1]

∣∣∣∣∣ 1

N∑n=1

1[0,a]({xn})− a

∣∣∣∣∣ .is called the star-discrepancy of x1, . . . , xN .

D∗N(x1, . . . , xN)→ 0.

Generalization to multidimensional case: consider axis-parallel boxes instead of

intervals.

Definition

D∗N(x1, . . . , xN) = supa∈[0,1]

∣∣∣∣∣ 1

N∑n=1

1[0,a]({xn})− a

D∗N(x1, . . . , xN)→ 0.

intervals.

Definition

D∗N(x1, . . . , xN) = supa∈[0,1]

∣∣∣∣∣ 1

N∑n=1

1[0,a]({xn})− a

D∗N(x1, . . . , xN)→ 0.

intervals.

Highlights and open problems II

Theorem (Koksma, 1935)

For almost all α > 1, the sequence (αn)n≥1 is u.d. mod 1.

Not a single specific number α is known for which the theorem holds.

How about α = 3/2? (Related to Waring’s problem)

Highlights and open problems III

A real number α is a normal number in base β if and only if

(βnα)n≥1 is u.d. mod 1.

Almost all numbers are normal. (Borel, 1909)

Open question: Are numbers such as√

2, π, e, . . . normalnumbers? (In base 2, base 10, etc.)

Highlights and open problems IV

Theorem (Ternary Goldbach conjecture; Vinogradov, 1937, andHelfgott, 2013)

Every odd number greater than 5 can be expressed as the sum ofthree primes.

Key ingredient: the sequence (pnα)n≥1 is u.d. mod 1 forirrational α.

Highlights and open problems V

Theorem (Erdos–Turan inequality, 1948)

For numbers x1, . . . , xN and an positive integer H we have

D∗N(x1, . . . , xN) ≤ cabs

H∑h=1

∣∣∣∣∣ 1

N∑n=1

e2πihxn

∣∣∣∣∣).

Highlights and open problems VI

Theorem (Koksma–Hlawka inequality; Hlawka, 1961)

Let f be a function on [0, 1]d which has bounded variation, and letx1, . . . , xN be points in [0, 1]d . Then∣∣∣∣∣

∫[0,1]d

f (x) dx − 1

N∑n=1

f (xn)

∣∣∣∣∣ ≤ (Var f ) · D∗N(x1, . . . , xN).

⇒ Fundamental result for Quasi-Monte Carlo integration.

Highlights and open problems VI

Theorem (Koksma–Hlawka inequality; Hlawka, 1961)

Let f be a function on [0, 1]d which has bounded variation, and letx1, . . . , xN be points in [0, 1]d . Then∣∣∣∣∣

∫[0,1]d

f (x) dx − 1

N∑n=1

f (xn)

∣∣∣∣∣ ≤ (Var f ) · D∗N(x1, . . . , xN).

⇒ Fundamental result for Quasi-Monte Carlo integration.

Highlights and open problems VII

Theorem (Halton; Hammersley; 1960)

There exist points x1, . . . , xN ∈ [0, 1]d such that

D∗N(x1, . . . , xN) ≤ cd(logN)d−1

Highlights and open problems VIII

Theorem (Roth, 1954)

Let d ≥ 3. Then for any set x1, . . . , xN of N points in [0, 1]d wehave

D∗N(x1, . . . , xN) ≥ cd(logN)

d−12

Highlights and open problems VIII

Theorem (Bilyk–Lacey–Vagharshakyan, 2008)

Let d ≥ 3. Then for any set x1, . . . , xN of N points in [0, 1]d wehave

D∗N(x1, . . . , xN) ≥ cd(logN)

d−12

⇒ Theory of irregularities of distributions.

Highlights and open problems IX

Theorem (R.C. Baker, 1980)

Let (an)n≥1 a strictly increasing sequence of positive integers.Then

D∗N(a1x , . . . , aNx) = O

((logN)3/2+ε√

)for almost all x .

The optimal exponent of the logarithmic term is an open problem.

⇒ Almost everywhere convergence of Fourier series, Carleson’s theorem.

Highlights and open problems IX

Theorem (R.C. Baker, 1980)

Let (an)n≥1 a strictly increasing sequence of positive integers.Then

D∗N(a1x , . . . , aNx) = O

((logN)3/2+ε√

)for almost all x .

The optimal exponent of the logarithmic term is an open problem.

⇒ Almost everywhere convergence of Fourier series, Carleson’s theorem.

Highlights and open problems X

Theorem (Hlawka, 1975)

The sequence of imaginary parts of the nontrivial zeroes of theRiemann zeta function, sorted in increasing order, is u.d. mod 1.

Highlights and open problems XI

Theorem (Heinrich–Novak–Wasilkowski–Wozniakowski, 2001)

There exits an absolute constant cabs such that the following holds.

For every ε ∈ (0, 1) and every d ≥ 1 there is a set of N pointsx1, . . . , xN ∈ [0, 1]d such that

D∗N(x1, . . . , xN) ≤ ε

andN ≤ cabsdε

In the upper bound for N, the dependence on d is optimal. The optimal

exponent of ε must be between −2 and −1 (Hinrichs, 2004).

⇒ Information-based complexity, tractability theory.

D∗N(x1, . . . , xN) ≤ ε

andN ≤ cabsdε

D∗N(x1, . . . , xN) ≤ ε

andN ≤ cabsdε

Thanks

Thank you!

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