Weyl's equidistribution theorem

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Problem I.3.9 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section I.3 "Irrationality"

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Let $f \in C(\left[0,1\right])$ and $x \in \mathbb {R} \backslash \mathbb {Q}$ . Prove that

\frac{1}{n} \sum _{1 \leqslant k \leqslant n} f(\delta (k x)) \rightarrow \int _{0}^{1} f(t) d t \quad \text{ as } \quad n \rightarrow +\infty .

Problem IX.4.11 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section IX.4 "Fourier Series and Transform"

(a)(b)

For which $p>0$ do the series

(a)

$\sum \frac{\sqrt[3]{\cos (2 \pi n \sqrt{2})}}{n^{p}}$ ;

(b)

$\sum (-1)^{n} \frac{\left|\sin (2 \pi n \sqrt{2})\right|}{n^{P}}$

converge?

*) For a proof not using Fourier series, see the hint to problem I.2.40. Various generalizations of the Wirtinger inequality can be found in [KhLP] on pp. 221, 384.

Problem IX.4.12 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section IX.4 "Fourier Series and Transform"

(a)(b)

Prove the convergence of the series

(a)

$\sum \frac{\sqrt[3]{\cos \left(\pi \sqrt{2} n^{2}\right)}}{n}$ ;

(b)

$\sum (-1)^{n} \frac{\left|\sin \left(\pi \sqrt{2} n^{3}\right)\right|}{n}$ .

Problem X.2.29 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.2 "Measure-Preserving Transformations"

(a)(b)

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$ . A set $A \subset X$ is called invariant if $T^{-1}(A)=A$ . A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$ , or $\mu (X \backslash A)=0$ .

Consider the irrational shift

S(x)=(x+\alpha ) \bmod 1 \quad (x \in [0,1)=X, \alpha \in \mathbb {R} \backslash \mathbb {Q}) .

It is ergodic by 2.21a). By the ergodic theorem, for any summable function $f$ on $X$ and for almost all $x \in X$ the equality

\lim \frac{1}{n} \sum _{0 \leqslant k<n} f\left(S^{k}(x)\right)=\int _{0}^{1} f(t) d t \tag {1}

holds.

(a)

Prove that if $f \in C(\left[0,1\right])$ , then for the shift $S$ one can assert more, namely that equality (1) holds for every $x \in [0,1)$ . (First assume that $f(0)=f(1)$ , and then consider the general case.)

(b)

Prove that if $\Delta =\langle a, b\rangle \subset [0,1), \chi_{\Delta }(x)=1$ for $x \in \Delta$ , $\chi_{\Delta }(x)=0$ for $x \notin \Delta$ , then

\lim \frac{1}{n} \sum _{0 \leqslant k<n} \chi _{\Delta }\left(S^{k}(x)\right)=b-a \tag {2}

for all $x \in [0,1)$ .

If $\left\{ S^{k}(x)\right\}$ is replaced by an arbitrary sequence $\left\{ x_{k}\right\}$ , then property b) is called uniform distribution of this sequence (see [PS], [KN]).

For a uniformly distributed sequence $\left\{ x_{k}\right\}$ and a function $f$ continuous on $\left[0,1\right]$ , a relation analogous to (1) holds:

\frac{1}{n} \sum _{k=1}^{n} f\left(x_{k}\right) \underset {n \rightarrow \infty }{\longrightarrow } \int _{0}^{1} f(t) d t

In particular,

\frac{1}{n} \sum _{k=1}^{n} e^{2 \pi i m x_{k}} \underset {n \rightarrow \infty }{\longrightarrow } 0 \text{ for any integer } m \neq 0 \text{. }

This condition is not only necessary but also sufficient for uniform distribution. The latter statement is called Weyl's criterion (cf. problem I.3.9).

Remark.

The Birkhoff--Khinchin ergodic theorem (see [Bi], [KSF]): let $T$ be a transformation of a set $X$ preserving a probability measure $\mu$ , and let $f$ be a measurable function such that $\int_{X}\left|f(x)\right| d \mu (x)<+\infty$ . Then for almost every (with respect to the measure $\mu$ ) $x \in X$ the limit

\lim \frac{1}{n} \sum _{0 \leqslant k<n} f\left(T^{k}(x)\right)=\bar{f}(x)

exists, called the mean value of the function $f$ along the trajectory of the point $x$ . Moreover the function $\bar{f}$ is invariant and $\int_{X} \bar{f}(x) d \mu (x)=\int_{X} f(x) d \mu (x)$ . If $T$ is ergodic, then $\bar{f}$ turns out to be a constant (see problem 2.20), equal to $\int_{X} f(x) d \mu (x)$ (i.e. the mean value of $f$ on $X$ ).

Problem 2.165 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Criteria of Equidistribution"

A sequence $x_{1}, x_{2}, x_{3}, \ldots , x_{n}, \ldots$ , $0 \leq x_{n} \leq 1$ , is equidistributed on $\left[0,1\right]$ if and only if the two equations

\lim _{n \to \infty } \frac{\cos 2\pi k x_{1} + \cos 2\pi k x_{2} + \cdots + \cos 2\pi k x_{n}}{n} = 0,

\lim _{n \to \infty } \frac{\sin 2\pi k x_{1} + \sin 2\pi k x_{2} + \cdots + \sin 2\pi k x_{n}}{n} = 0

hold for every positive integer $k$ .

Remark.

Problem 137.

Problem 2.166 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Multiples of an Irrational Number"

Let $\theta$ be an irrational number. The numbers

x_{n} = \theta n - \left[\theta n\right]

are equidistributed on the interval $\left[0,1\right]$ .

Problem 2.167 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Multiples of an Irrational Number"

Let $\theta$ denote an irrational number. Put $\varepsilon_{n}=1$ or $\varepsilon_{n}=0$ according as the integer next to $n\theta$ is larger or smaller than $n\theta$ . Let $a$ and $d$ , $a \geq 0$ , $d > 0$ , stand for two integers. Then we find that

\lim _{n \to \infty } \frac{\sum _{k=0}^{n-1} \varepsilon _{a+kd}}{n} = \frac{1}{2} .

Problem 2.173 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Multiples of an Irrational Number"

Let $\theta$ be an irrational number, $x_{n} = n\theta - \left[n\theta \right]$ , $n = 1, 2, 3, \ldots$ and let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots , \alpha_{n}, \ldots$ be a monotone decreasing sequence of positive numbers whose sum diverges. Then we find for any properly integrable function $f(x)$ on $\left[0,1\right]$ that

\lim _{n \to \infty } \frac{\alpha _{1}f(x_{1}) + \alpha _{2}f(x_{2}) + \cdots + \alpha _{n}f(x_{n})}{\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}} = \int _{0}^{1} f(x) \, dx .

Problem 1.8.19 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.8 "Some Special Functions"

Suppose $f$ is a continuous function on $\mathbb {R}^{1}, f(x+2 \pi ) = f(x)$ , and $\alpha / \pi$ is irrational. Prove that

\lim _{N \rightarrow \infty } \frac{1}{N} \sum _{n = 1}^{N} f(x+n \alpha ) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } f(t) d t

for every $x$ . Hint: Do it first for $f(x) = e^{i k x}$ .

Problem 2.4.13 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.4 "Elementary Hilbert Space Theory"

Suppose $f$ is a continuous function on $\mathbb {R}^{1}$ , with period 1. Prove that

\lim _{N \rightarrow \infty } \frac{1}{N} \sum _{n = 1}^{N} f(n \alpha ) = \int _{0}^{1} f(t) d t

for every irrational real number $\alpha$ . Hint : Do it first for

f(t) = \exp (2 \pi i k t), \quad k = 0, \pm 1, \pm 2, \ldots

Problem 9.34 from Athreya K.B., Lahiri S.N. "Measure Theory and Probability Theory" [AtLa06]More problems in Chapter 9 "Convergence in Distribution"

(Weyl's equi-distribution property). Let $0 < \alpha < 1$ be an irrational number. Let $\mu_{n}(\cdot )$ be the measure defined by $\mu_{n}(A) \equiv \frac{1}{n} \sum_{j = 0}^{n-1} I_{A}(j \alpha \bmod 1), A \in \mathcal{B}([0,1])$ . Show that $\mu_{n} \longrightarrow^{d}$ Uniform $(0,1)$ .

Problem 9.7.15 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 9.7 "Problems"

Let $g: \mathbb {R} \rightarrow \mathbb {R}$ be periodic with period $1$ , and uniformly continuous and integrable over $[0,1]$ . Define $Z_{n} = g(X+(n-1) \alpha ), n \geq 1$ , where $X$ is uniform on $[0,1]$ and $\alpha$ is irrational. Show that, as $n \rightarrow \infty$ ,

\frac{1}{n} \sum _{j = 1}^{n} Z_{j} \rightarrow \int _{0}^{1} g(u) d u \quad \text{ a.s. }

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[Rud76] Rudin W. Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[AtLa06] Athreya K.B., Lahiri S.N. Measure Theory and Probability Theory. Springer, 2006.

[GrSt20] Grimmett G.R., Stirzaker D.R. Probability and Random Processes. 4th ed., Oxford University Press, 2020.