Math Problems Vault

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.

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(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.

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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 X.2.30 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)

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(a)

Two points move along a circle, jumping once a second by the same angle, which is incommensurable with $\pi$ . What fraction of a sufficiently long time interval will both points simultaneously lie on some (fixed in advance) semicircle, if at the initial moment the angle between the points was equal to $\alpha$ ?

(b)

Answer the same question if every second one point moves by an angle $\beta_{1}$ , and the other by an angle $\beta_{2}$ , where the numbers $\beta_{1}, \beta_{2}$ , $\pi$ are linearly independent over $\mathbb {Q}$ (see the definition at the beginning of 4 of Chapter VII).

Problem X.2.31 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"

Let $x=\sum \frac{x_{k}}{2^{k}}, x_{k} \in \left\{ 0,1\right\}$ be the binary expansion of a number $x \in [0,1)$ . Using the ergodicity of the doubling transformation (see problem 2.21b)), prove that for almost all $x \in [0,1)$ the limit $\lim \frac{1}{n} \sum_{0 \leqslant k<n} x_{k}$ exists (the frequency of occurrence of the digit one), and find it. Solve the analogous problem for expansions in a base- $p$ fraction (cf. problems IX.2.14, IX.2.16).

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Problem X.2.35 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"

Let $a_{k}$ be the first digit of the decimal representation of the number $2^{k} (k=0,1, \ldots )$ , i.e., $a_{0}=1, a_{1}=2, a_{2}=4, a_{3}=8, a_{4}=1, \ldots$ What is the frequency of occurrence of the digit $p \in \left\{ 1, \ldots , 9\right\}$ in this sequence?

As is well known, every number $x$ from the interval $(0,1]$ can be uniquely represented as a continued fraction

x=\frac{1}{a_{1}+\frac{1}{a_{2}+\cdots }}

which is also written as $x=\left[a_{1}, a_{2}, \ldots \right]$ , where $a_{1}=a_{1}(x), a_{2}=a_{2}(x)$ , $a_{3}=a_{3}(x), \ldots$ are natural numbers. Such a fraction contains finitely many terms only when $x$ is rational. The finite continued fraction $\left[a_{1}, \ldots , a_{n}\right]$ is uniquely written as an irreducible fraction $\frac{p_{n}}{q_{n}}$ . The value of an infinite continued fraction is understood as the limit of the finite ones:

\left[a_{1}, a_{2}, \ldots \right]=\lim \left[a_{1}, \ldots , a_{n}\right]=\lim \frac{p_{n}}{q_{n}},

the number $\frac{p_{n}}{q_{n}}=\frac{p_{n}(x)}{q_{n}(x)}$ is called the convergent. It is easily verified that

\left|x-\frac{p_{n}(x)}{q_{n}(x)}\right| \leqslant \frac{1}{q_{n}(x) q_{n+1}(x)}

A variety of information about continued fractions can be gleaned from the book $[\mathrm{X}]$ . Associated with the expansion of numbers into continued fractions is the Gauss transformation

T(x)=\frac{1}{x} \bmod 1(x \in (0,1]) .

Indeed, let $x=\left[a_{1}, a_{2}, \ldots \right], y=\left[a_{2}, a_{3}, \ldots \right]$ . Then $x=\frac{1}{a_{1}+y}$ , from which it follows that $y=T(x)$ . Using $T$ , the numbers $a_{n}$ are written as follows:

a_{1}(x)=\left[\frac{1}{x}\right], a_{2}(x)=a_{1}(T(x)), \ldots , a_{n}(x)=a_{1}\left(T^{n-1}(x)\right) .

The Gauss transformation preserves the measure $d \mu =\frac{c}{1+x} d x$ (see problem 2.5). The measure $\mu$ is a probability measure for $c=\frac{1}{\ln 2}$ . The reader can learn about interesting applications of ergodic theory to number theory and continued fractions in [Po].

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Digit and orbit frequencies via the Birkhoff theorem

References