Almost Periodic Functions and Sequences

Prove that if $\lambda$ is an irrational number, then for any $a \in [0,1)$ and $\varepsilon >0$ there exists $n \in \mathbb {Z}$ such that $n \lambda \stackrel{\varepsilon }{=} a(\bmod 1)$. Verify that the set of such $n$ is relatively dense on the line (compare with problem I.3.2).

a) Prove that if $\lambda_{1}, \lambda_{2} \neq 0$ and the ratio $\lambda_{1} / \lambda_{2}$ is irrational, then for any $a_{1}, a_{2} \in [0,1)$ and $\varepsilon >0$ there exists $t \in \mathbb {R}$ such that

Conversely, if system (1) is solvable for $t$ for any $a_{1}$, $a_{2}, \varepsilon$, then the ratio $\lambda_{1} / \lambda_{2}$ is irrational. b) Prove that if $\lambda_{1}$ and $\lambda_{2}$ are independent over $\mathbb {Q}$ modulo 1, then there exists a relatively dense set of integer solutions of system (1) on the line.

What condition must the real numbers $\lambda_{1}, \ldots , \lambda_{k}$ satisfy so that for any $\varepsilon >0$ the system of equations

has a solution $t$ satisfying $\left|t\right|>\left|\lambda_{j}\right|^{-1}$ for some $j$?

What condition must the real numbers $\lambda_{1}, \ldots , \lambda_{k}$ satisfy so that for any $a_{1}, \ldots , a_{k} \in [0,1)$ and $\varepsilon >0$ the system of equations

has

What condition must the real numbers $\lambda_{1}, \ldots , \lambda_{k}$ satisfy so that for any $a_{1}, \ldots , a_{k} \in [0,1)$ and $\varepsilon >0$ the system of equations

has

Let $f(x)=\sin (a x+b)+\sin (c x+d), x \in \mathbb {R}$, where $a c \neq 0$. Prove that if the function $f$ is not identically zero, then it takes values of different signs.

Prove that every quasi-trigonometric polynomial

possesses the property of uniform almost periodicity. Verify that the polynomial $f$ is periodic if and only if the factors $\lambda_{1}, \ldots , \lambda_{n}$ are pairwise commensurable (i.e. their ratios are rational).

Let $f \in C\left(\mathbb {R}^{n}\right)$ be a function having period 1 in each argument, $\alpha_{1}, \ldots , \alpha_{n} \in \mathbb {R}$ numbers linearly independent over $\mathbb {Q}$, $\beta_{1}, \ldots , \beta_{n} \in \mathbb {R}$. Prove that the function

is uniformly almost periodic.

Prove that if $f$ is a continuous u.a.p. function and its $\varepsilon$-almost periods are bounded for small $\varepsilon$, then $f$ is periodic.

Prove that a u.a.p.function continuous on $\mathbb {R}$ is uniformly continuous on the line and bounded. Do unbounded nonperiodic u.a.p.functions exist? Does continuity follow from boundedness?

Prove that the limit of a uniformly convergent (on $\mathbb {R}$) sequence of u.a.p.functions is a u.a.p.function.

Prove that if a u.a.p.function has a derivative that exists and is uniformly continuous, then it is also a u.a.p.function.

Prove that if the antiderivative $F$ of a continuous u.a.p.function $f$ is bounded, then it is also a u.a.p.function.

Let $f \in C(\mathbb {R}), h \in \mathbb {R}, f_{h}(x)=f(x+h)$. Prove that the function $f$ has the u.a.p.property if and only if, for every numerical sequence $\left\{ h_{n}\right\}$, one can choose from the sequence $\left\{ f_{h_{n}}\right\}$ a subsequence that converges uniformly on $\mathbb {R}$ (compactness of the family of shifts).

Prove that the sum and product of continuous u.a.p.functions are again a u.a.p.function.

Prove that for every u.a.p.function $f$ there exists a mean value

Let $\langle f, g\rangle =M(f \bar{g})$, where $f, g$ are continuous u.a.p.functions, and the functional $M$ is defined in the previous problem. Prove that

Let $\left\{ a_{n}\right\} , a_{n} \in \left\{ 0,1,2\right\}$, be a nonperiodic sequence. Define the sequences $\left\{ b_{n}\right\}$ and $\left\{ c_{n}\right\}$ as follows:

Prove that at least one of the sequences $\left\{ b_{n}\right\} ,\left\{ c_{n}\right\}$ is nonperiodic.

Let $\left\{ \varepsilon_{k}\right\}_{k \in \mathbb {Z}}$ be an almost periodic sequence, $f(x)=\varepsilon_{\left[x\right]}(x \in \mathbb {R},\left[x\right]$ the integer part of the number $x)$. Is $f$ a u.a.p.function?

Is it true that for every almost periodic sequence $\left\{ \varepsilon_{k}\right\}_{k \in \mathbb {Z}}$ there exists a limit of the Cesàro means $\lim \frac{1}{2 n+1} \sum_{-n \leqslant k \leqslant n} \varepsilon_{k} ?$

Let $k>0$ be an irrational number, $S=\left\{ (x, y) \mid k x \leqslant y<k(x+1)+1\right\}$ a strip in $\mathbb {R}^{2}$, and $\Sigma =S \cap \mathbb {Z}^{2}$ the set of integer points in $S$. Verify that the orthogonal projections of the points of $\Sigma$ onto the side of the strip (Fig. 10) divide it into segments whose lengths take only two values: $a$ and $b$. Numbering the segments by integers in the order of their location on the line, set $\varepsilon_{i}=0$ if the $i$-th segment has length $a$, and $\varepsilon_{i}=1$ otherwise. Prove that the sequence $\left\{ \varepsilon_{i}\right\}_{i \in \mathbb {Z}}$ is almost periodic.

Fig. 10

Consider the Morse sequence

defined by the equalities $\varepsilon_{1}=0, \quad \varepsilon_{2}=1, \quad \varepsilon_{k}=1-\varepsilon_{k-2^{n}} \quad$ for $2^{n}<k \leqslant 2^{n+1}(n \in \mathbb {N})$. Prove that it is almost periodic.

For a finite sequence ("word")

of zeros and ones, let $A^{0}$ denote $A$, and $A^{1}=\bar{\varepsilon }_{1} \bar{\varepsilon }_{2} \ldots \bar{\varepsilon }_{n}$, where $\bar{\varepsilon }_{k}=1-\varepsilon_{k}$. Define a sequence $\varepsilon$ by specifying its initial segments $A_{p}$, which are defined recursively: $A_{0}$ is any word, $A_{0}=\varepsilon_{1} \varepsilon_{2} \ldots \varepsilon_{n}$,

where $\xi_{1} \ldots \xi_{l}$ is some word of zeros and ones,

and so on. Thus $A_{0}$ serves as the beginning of $A_{1}$, $A_{1}$ as the beginning of $A_{2}$, and so on. The infinite sequence $\left\{ A_{p}\right\}$ defines a certain sequence $\varepsilon$. Prove that it is periodic or almost periodic.

A folding sequence is a sequence $s=\left\{ s_{n}\right\}$ of zeros and ones whose initial segments $A_{n}$ of length $2 k-1\left(k=2^{n}\right)$ have the form

The name of the sequence is explained by the fact that the word $A_{n}$ can be associated with the sequence of folds of a paper strip formed on it as a result of folding it $(n+1)$ times (first in half, then in half again, and so on). If, moving along the strip, we assign a one to each fold resulting in a turn to the left, and a zero to a "turn right" fold, we obtain $A_{n}$ (Fig. 11). The sequence is uniquely determined by specifying the numbers $s_{2^{n}}, n=0,1,2, \ldots$

Fig. 11

Prove that every folding sequence is almost periodic but not periodic, even if we restrict to terms with arbitrarily large indices.

Prove that fold sequences (see Problem 4.24) have the following property (Besicovitch almost periodicity): for every $\varepsilon >0$ there exist a number $L>0$ and numbers $\tau_{m} \in \mathbb {N}(m \in \mathbb {N})$ such that

1. if $\nu (\Delta )$ is the number of those $\tau_{m}$ that fall into the interval $\Delta \subset \mathbb {R}_{+}$, then $v\left(\Delta_{1}\right)<2 v\left(\Delta_{2}\right)$ for all intervals $\Delta_{1}, \Delta_{2}$ of length $L$;

2. $\varlimsup_{n} \frac{1}{n} \sum_{1 \leqslant j \leqslant n}\left|s_{j+\tau_{m}}-s_{j}\right|<\varepsilon$ for every $m \in \mathbb {N}$;

3. $\varlimsup_{n} \varlimsup_{p} \frac{1}{n p} \sum_{1 \leqslant j \leqslant n} \sum_{1 \leqslant m \leqslant p}\left|s_{j+\tau_{m}}-s_{j}\right|<\varepsilon$.

Prove that if $x_{n}=(-1)^{s_{n}}, n \in \mathbb {N}$, where $\left\{ s_{n}\right\}$ is a fold sequence, then for any $t \in \mathbb {R}$ the limit

exists (the "Fourier coefficient"), and moreover

and $c_{t}=0$ otherwise. Verify that the "Parseval equality" holds:

Consider the binary sequence

(the Rudin-Shapiro sequence), whose initial segments $A_{n}$ of length $2^{n}$ have the form $A_{n}=C_{n} D_{n}$, where $C_{n}$ and $D_{n}$ are words of length $2^{n-1}$, with $A_{1}=C_{1} D_{1}=00$, and for $n>1$ the words $A_{n}, C_{n}, D_{n}$ are defined recursively:

($D_{n-1}^{1}$ is obtained from $D_{n-1}$ by swapping $0 \leftrightarrow 1$). Prove the almost periodicity of this sequence.

Note that the sequence $\left\{ (-1)^{r_{n}}\right\}$ is the sequence of coefficients of the Rudin-Shapiro polynomials (see Problem IX.4.25).