Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
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
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Hermann Weyl
Hermann Weyl (1885–1955)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Hermann Weyl
David Hilbert and Hermann Weyl
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Hermann Weyl
Bust of Hermann Weyl at the ETH Zurich
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Hermann Weyl
Weyl riding a seesaw
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Life
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Work
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
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Work II
Topics named after Hermann Weyl (Wikipedia)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Fame
The Greatest Mathematicians of All Time
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Citations
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Section 2
Prehistory of uniform distribution theory
Christoph Aistleitner One Hundred Years 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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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→∞
1
N
N∑n=1
1[0,a]({xn}) = a
for all a ∈ [0, 1].
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Section 3
Weyl’s paper of 1916
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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→∞
1
N
N∑n=1
f (xn) =
∫ 1
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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→∞
1
N
N∑n=1
f (xn) =
∫ 1
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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→∞
1
N
N∑n=1
f (xn) =
∫ 1
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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→∞
1
N
N∑n=1
f (xn) =
∫ 1
0f (x) dx .
holds.
⇒ Hidden idea 3: Topological definition – assume that f is continuous.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
“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→∞
1
N
N∑n=1
e2πihxn = 0
holds for all integers h 6= 0.
⇒ Establishes connection between uniform distribution theory and exponential
sums.
Hardy–Littlewood: “The proof depends on a simple but ingenious use of the
theory of approximation to arbitrary functions by finite trigonometric
polynomials.”
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
“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→∞
1
N
N∑n=1
e2πihxn = 0
holds for all integers h 6= 0.
⇒ Establishes connection between uniform distribution theory and exponential
sums.
Hardy–Littlewood: “The proof depends on a simple but ingenious use of the
theory of approximation to arbitrary functions by finite trigonometric
polynomials.”
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
“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→∞
1
N
N∑n=1
e2πihxn = 0
holds for all integers h 6= 0.
⇒ Establishes connection between uniform distribution theory and exponential
sums.
Hardy–Littlewood: “The proof depends on a simple but ingenious use of the
theory of approximation to arbitrary functions by finite trigonometric
polynomials.”
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
“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→∞
1
N
N∑n=1
e2πihxn = 0
holds for all integers h 6= 0.
⇒ Establishes connection between uniform distribution theory and exponential
sums.
Hardy–Littlewood: “The proof depends on a simple but ingenious use of the
theory of approximation to arbitrary functions by finite trigonometric
polynomials.”
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Proof that (nα)n≥1 is u.d. mod 1 for irrational α:
We have
1
N
N∑n=1
e2πihnα =1
Ne2πihα
1− e2πihNα
1− e2πihα→ 0,
since hα 6∈ Z.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Proof that (nα)n≥1 is u.d. mod 1 for irrational α:
We have
1
N
N∑n=1
e2πihnα =1
Ne2πihα
1− e2πihNα
1− e2πihα→ 0,
since hα 6∈ Z.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
“We remark that whenever the limit equation
limN→∞
1
N
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?
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
“We remark that whenever the limit equation
limN→∞
1
N
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?
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional theory
Continuous uniform distribution
Interpretation in terms of probability theory
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional theory
Continuous uniform distribution
Interpretation in terms of probability theory
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional theory
Continuous uniform distribution
Interpretation in terms of probability theory
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
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).
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
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).
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional polynomial sequences
Characterization of multidimensional uniform distribution
Connections with geometry of numbers
Sequences with multi-indices
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional polynomial sequences
Characterization of multidimensional uniform distribution
Connections with geometry of numbers
Sequences with multi-indices
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional polynomial sequences
Characterization of multidimensional uniform distribution
Connections with geometry of numbers
Sequences with multi-indices
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Multidimensional polynomial sequences
Characterization of multidimensional uniform distribution
Connections with geometry of numbers
Sequences with multi-indices
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Appendix on compact manifolds
⇒ Hidden idea: Uniform distribution in compact spaces and uniform
distribution in topological groups.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Weyl’s paper of 1916
Appendix on compact manifolds
⇒ Hidden idea: Uniform distribution in compact spaces and uniform
distribution in topological groups.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Section 4
Highlights and open problems
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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∑n=1
1[0,a]({xn})− a
∣∣∣∣∣ .is called the star-discrepancy of x1, . . . , xN .
A sequence (xn)n≥1 is u.d. mod 1 if and only if
D∗N(x1, . . . , xN)→ 0.
Generalization to multidimensional case: consider axis-parallel boxes instead of
intervals.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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∑n=1
1[0,a]({xn})− a
∣∣∣∣∣ .is called the star-discrepancy of x1, . . . , xN .
A sequence (xn)n≥1 is u.d. mod 1 if and only if
D∗N(x1, . . . , xN)→ 0.
Generalization to multidimensional case: consider axis-parallel boxes instead of
intervals.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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∑n=1
1[0,a]({xn})− a
∣∣∣∣∣ .is called the star-discrepancy of x1, . . . , xN .
A sequence (xn)n≥1 is u.d. mod 1 if and only if
D∗N(x1, . . . , xN)→ 0.
Generalization to multidimensional case: consider axis-parallel boxes instead of
intervals.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.)
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
(1
H+
H∑h=1
1
h
∣∣∣∣∣ 1
N
N∑n=1
e2πihxn
∣∣∣∣∣).
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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∑n=1
f (xn)
∣∣∣∣∣ ≤ (Var f ) · D∗N(x1, . . . , xN).
⇒ Fundamental result for Quasi-Monte Carlo integration.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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∑n=1
f (xn)
∣∣∣∣∣ ≤ (Var f ) · D∗N(x1, . . . , xN).
⇒ Fundamental result for Quasi-Monte Carlo integration.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
N.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
N.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
+εd
N.
⇒ Theory of irregularities of distributions.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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√
N
)for almost all x .
The optimal exponent of the logarithmic term is an open problem.
⇒ Almost everywhere convergence of Fourier series, Carleson’s theorem.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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√
N
)for almost all x .
The optimal exponent of the logarithmic term is an open problem.
⇒ Almost everywhere convergence of Fourier series, Carleson’s theorem.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
−2.
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
−2.
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
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
−2.
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.
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Thanks
Thank you!
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory
Christoph Aistleitner One Hundred Years of Uniform Distribution Theory