Metric number theory, lacunary series and systems of dilated - - PowerPoint PPT Presentation

Metric number theory, lacunary series and systems

• f dilated functions

Christoph Aistleitner

aistleitner@math.tugraz.at

Kobe University, Japan

Uniform distribution and quasi-Monte Carlo methods RICAM Linz, Austria

October 14, 2013

The equidistribution theorem

The sequence of fractional parts ({kx})k≥1 is uniformly distributed modulo 1 if and only if x ∈ Q (Bohl–Sierpinski–Weyl, 1909–1910). A sequence (xk)k≥1 is called uniformly distributed modulo 1 if for all a ∈ [0, 1] the relation lim

N→∞

1 N

N

• k=1

1[0,a)(xk) = a holds.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

The equidistribution theorem

The sequence of fractional parts ({kx})k≥1 is uniformly distributed modulo 1 if and only if x ∈ Q (Bohl–Sierpinski–Weyl, 1909–1910). A sequence (xk)k≥1 is called uniformly distributed modulo 1 if for all a ∈ [0, 1] the relation lim

N→∞

1 N

N

• k=1

1[0,a)(xk) = a holds.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

The equidistribution theorem

For a sequence (nk)k≥1 of distinct integers, the sequence of fractional parts ({nkx})k≥1 is uniformly distributed modulo 1 for almost all x. (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for nk = 2k, k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

The equidistribution theorem

For a sequence (nk)k≥1 of distinct integers, the sequence of fractional parts ({nkx})k≥1 is uniformly distributed modulo 1 for almost all x. (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for nk = 2k, k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

The equidistribution theorem

For a sequence (nk)k≥1 of distinct integers, the sequence of fractional parts ({nkx})k≥1 is uniformly distributed modulo 1 for almost all x. (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for nk = 2k, k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

The equidistribution theorem

For a sequence (nk)k≥1 of distinct integers, the sequence of fractional parts ({nkx})k≥1 is uniformly distributed modulo 1 for almost all x. (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for nk = 2k, k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Discrepancy

For a finite sequence x1, . . . , xN of points in the unit interval the star-discrepancy D∗

N is defined as

D∗

N(x1, . . . , xN) = sup 0≤a≤1

• 1

N

N

• k=1

1[0,a)(xk) − a

• .

We want to find upper bounds for D∗

N({n1x}, . . . , {nNx})

as N → ∞.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Discrepancy

For a finite sequence x1, . . . , xN of points in the unit interval the star-discrepancy D∗

N is defined as

D∗

N(x1, . . . , xN) = sup 0≤a≤1

• 1

N

N

• k=1

1[0,a)(xk) − a

• .

We want to find upper bounds for D∗

N({n1x}, . . . , {nNx})

as N → ∞.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Closely related problems

Find upper bounds for

• N
• k=1

1[0,a)({nkx}) − Na

• for fixed a and a.e. x.

Find upper bounds for

• N
• k=1

f (nkx)

• for f being centered, 1-periodic and of bounded variation, for

a.e. x. Find criteria for the a.e. convergence of

∞

• k=1

ckf (nkx) for f being centered, 1-periodic and of bounded variation.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Closely related problems

Find upper bounds for

• N
• k=1

1[0,a)({nkx}) − Na

• for fixed a and a.e. x.

Find upper bounds for

• N
• k=1

f (nkx)

• for f being centered, 1-periodic and of bounded variation, for

a.e. x. Find criteria for the a.e. convergence of

∞

• k=1

ckf (nkx) for f being centered, 1-periodic and of bounded variation.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Closely related problems

Find upper bounds for

• N
• k=1

1[0,a)({nkx}) − Na

• for fixed a and a.e. x.

Find upper bounds for

• N
• k=1

f (nkx)

• for f being centered, 1-periodic and of bounded variation, for

a.e. x. Find criteria for the a.e. convergence of

∞

• k=1

ckf (nkx) for f being centered, 1-periodic and of bounded variation.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks I: Known results

In a few special cases the answers are known. If nk = k, k ≥ 1 (connection with continued fractions expansion). If (nk)k≥1 is increasing very quickly (lacunary sequences - almost independent behavior).

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks I: Known results

In a few special cases the answers are known. If nk = k, k ≥ 1 (connection with continued fractions expansion). If (nk)k≥1 is increasing very quickly (lacunary sequences - almost independent behavior).

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks I: Known results

In a few special cases the answers are known. If nk = k, k ≥ 1 (connection with continued fractions expansion). If (nk)k≥1 is increasing very quickly (lacunary sequences - almost independent behavior).

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks II: Probabilistic interpretation

It suffices to consider x ∈ [0, 1]. The space ([0, 1], B, λ) is a probability space. The functions {nkx} and f (nkx) are random variables. For each k, the random variable {nkx} has distribution U([0, 1]). The discrepancy is the Kolmogorov–Smirnov statistic.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks II: Probabilistic interpretation

It suffices to consider x ∈ [0, 1]. The space ([0, 1], B, λ) is a probability space. The functions {nkx} and f (nkx) are random variables. For each k, the random variable {nkx} has distribution U([0, 1]). The discrepancy is the Kolmogorov–Smirnov statistic.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks II: Probabilistic interpretation

It suffices to consider x ∈ [0, 1]. The space ([0, 1], B, λ) is a probability space. The functions {nkx} and f (nkx) are random variables. For each k, the random variable {nkx} has distribution U([0, 1]). The discrepancy is the Kolmogorov–Smirnov statistic.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks II: Probabilistic interpretation

It suffices to consider x ∈ [0, 1]. The space ([0, 1], B, λ) is a probability space. The functions {nkx} and f (nkx) are random variables. For each k, the random variable {nkx} has distribution U([0, 1]). The discrepancy is the Kolmogorov–Smirnov statistic.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Remarks II: Probabilistic interpretation

It suffices to consider x ∈ [0, 1]. The space ([0, 1], B, λ) is a probability space. The functions {nkx} and f (nkx) are random variables. For each k, the random variable {nkx} has distribution U([0, 1]). The discrepancy is the Kolmogorov–Smirnov statistic.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Tools I

Erd˝

• s–Tur´

an inequality: For any positive integer H D∗

N(x1, . . . , xN) ≤ 3

H + 3

H

• h=1

1 h

• 1

N

N

• k=1

e2πihxk

• .

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Tools II

Koksma’s inequality: For any function f which has bounded variation Var(f ) in the unit interval we have

• 1

N

N

• k=1

f (xk) − 1 f (x) dx

• ≤ Var(f ) · D∗

N(x1, . . . , xN).

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Tools III

Carleson’s theorem: The Fourier series of a function in L2([0, 1]) is a.e. convergent. Equivalently: there exists an absolute constant c such that for any f ∈ L2([0, 1]) S∗f 2 ≤ cf 2.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Tools III

Carleson’s theorem: The Fourier series of a function in L2([0, 1]) is a.e. convergent. Equivalently: there exists an absolute constant c such that for any f ∈ L2([0, 1]) S∗f 2 ≤ cf 2.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Tools IV

G´ al’s theorem: There is an absolute constant c such that for any distinct integers n1, . . . , nN we have

N

• k,l=1

(gcd(nk, nl))2 nknl ≤ cN(log log N)2.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

An upper bound for trigonometric sums

For any increasing sequence of integers (nk)k≥1 we have

• N
• k=1

cos(2πnkx)

• = O

√ N(log N)1/2+ε a.e. This upper bound is optimal (Berkes–Philipp, 1994). For the discrepancy we get D∗

N({n1x}, . . . , {nNx}) = O

• (log N)3/2+ε

√ N

• a.e.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

An upper bound for trigonometric sums

For any increasing sequence of integers (nk)k≥1 we have

• N
• k=1

cos(2πnkx)

• = O

√ N(log N)1/2+ε a.e. This upper bound is optimal (Berkes–Philipp, 1994). For the discrepancy we get D∗

N({n1x}, . . . , {nNx}) = O

• (log N)3/2+ε

√ N

• a.e.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

An upper bound for trigonometric sums

For any increasing sequence of integers (nk)k≥1 we have

• N
• k=1

cos(2πnkx)

• = O

√ N(log N)1/2+ε a.e. This upper bound is optimal (Berkes–Philipp, 1994). For the discrepancy we get D∗

N({n1x}, . . . , {nNx}) = O

• (log N)3/2+ε

√ N

• a.e.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

An upper bound for sums of dilated functions

How about sums f (nkx)? We have to estimate

• max

1≤M≤N

• M
• k=1

f (nkx)

• 2

.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

An upper bound for sums of dilated functions

How about sums f (nkx)? We have to estimate

• max

1≤M≤N

• M
• k=1

f (nkx)

• 2

.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

For f being centered, 1-period and of bounded variation, and for (nk)k≥1 being an increasing sequence of integers, we have

• N
• k=1

f (nkx)

• = O

√ N(log N)3/2+ε a.e.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

An upper bound for sums of dilated functions II

Theorem (A–Berkes–Seip, 2013, to appear in Journal of the EMS) Let f be a centered, 1-periodic function of bounded variation. Let (nk)k≥1 be an increasing sequence of positive integers. Then

• N
• k=1

f (nkx)

• = O

√ N(log N)1/2(log log N)5/2+ε a.e.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Theorem (A–Berkes–Seip, 2013) Let f be a centered, 1-periodic function of bounded variation, and let (nk)k≥1 be an increasing sequence of positive integers. Let (ck)k≥1 be a real sequence satisfying

∞

• k=3

c2

k(log log k)4+ε < ∞.

Then the series

∞

• k=1

ckf (nkx) is a.e. convergent.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Problems I

We can estimate

• max

1≤M≤N

• M
• k=1

f (nkx)

• 2

. For the discrepancy we need

• max

1≤M≤N

sup

a∈[0,1]

• M
• k=1

1[0,a)({nkx}) − Ma

• 2

.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Problems I

We can estimate

• max

1≤M≤N

• M
• k=1

f (nkx)

• 2

. For the discrepancy we need

• max

1≤M≤N

sup

a∈[0,1]

• M
• k=1

1[0,a)({nkx}) − Ma

• 2

.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Problems II

Investigate the a.e. convergence problem for

∞

• k=1

ckf (kx). Investigate connection with the Duffin–Schaeffer conjecture.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func

Problems II

Investigate the a.e. convergence problem for

∞

• k=1

ckf (kx). Investigate connection with the Duffin–Schaeffer conjecture.

Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func