Cantor series expansions of rational numbers

TL;DR

This survey explores the conditions under which rational numbers can be represented using Cantor series, providing necessary and sufficient criteria for arbitrary sequences.

Contribution

It offers a comprehensive set of necessary and sufficient conditions for representing rational numbers via Cantor series with any given sequence.

Findings

Established necessary and sufficient conditions for rational representations

Generalized conditions for arbitrary sequences $(q_k)$

Enhanced understanding of Cantor series representations

Abstract

This survey is devoted to necessary and suffcient conditions for a rational number to be representable by a Cantor series. Necessary and suffcient conditions are formulated for the case of an arbitrary sequence $(q_k)$.

Equations

$$\frac{{\epsilon }_1}{q_1}+\frac{{\epsilon }_2}{q_1{q}_2}+\cdots +\frac{{\epsilon }_k}{q_1{q}_2\dots q_k}+\dots$$

$$\frac{{\alpha }_1}{b}+\frac{{\alpha }_2}{b^2}+\cdots +\frac{{\alpha }_n}{b^n}+\dots$$

$${\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_{m-1}{\epsilon }_{m}000\dots }^Q={\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_{m-1}[{\epsilon }_m-1][q_{m+1}-1][q_{m+2}-1]\dots }^Q=\sum _{i=1}^m\frac{{\epsilon }_i}{q_1{q}_2\dots q_i}.$$

$$n=\sum _{k=1}^{\infty }{\epsilon }_k{q}_1{q}_2\dots q_k,$$

$$R_I(Q)=\left\{x:x=\sum _{k=1}^{\infty }\frac{{\epsilon }_k}{q_1{q}_2\dots q_k},{\epsilon }_k\in I_k\subseteq {\Theta }_k\right\}.$$

$$\underset{k\to \infty }{\lim }\frac{\log q_k}{\log q_1{q}_2\dots q_k}=0$$

$$u=f(x)=\sum _{k=1}^{\infty }\frac{u_k}{k(k+1)},$$

$$u_{k+1}=\{\begin{array}{ll}-\frac{u_k}{k},& \text{if εk+1=0 but εk=0,}\\ & \text{or if εk+1=qk+1−1 but εk=qk−1;}\\ u_k,& \text{otherwise.}\end{array}$$

$$\underset{k\to \infty }{\lim }\frac{q_1{q}_2\dots q_k}{k!}=\infty$$

$${\psi }_{P,Q}(x)=\sum _{k=1}^{\infty }\frac{\min (E_k,q_k-1)}{q_1{q}_2\dots q_k},$$

$$x=E_0+\sum _{k=1}^{\infty }\frac{E_k}{p_1{p}_2\dots p_k},{\epsilon }_0+\sum _{k=1}^{\infty }\frac{{\epsilon }_k}{q_1{q}_2\dots q_k}.$$

$$A_i=\frac{h}{k}(B_i-1)$$

$$x=X=A_0+\frac{A_1}{B_1}+\frac{A_2}{B_1{B}_2}+\cdots +\frac{A_n}{B_1{B}_2\cdots B_n}+\dots ,$$

$$B_1=q_1{q}_2\cdots q_{i_1},B_2=q_{i_1+1}{q}_{i_2+1}\cdots q_{i_1+i_2},\dots ,$$

$$\frac{{\epsilon }_1}{q_1}+\frac{{\epsilon }_2}{q_1{q}_2}+\cdots +\frac{{\epsilon }_{i_1}}{q_1{q}_2\cdots q_{i_1}}=\frac{A_1}{B_1}.$$

$$q_1{q}_2\dots q_{n_0}\equiv 0(\mathrm{mod}r).$$

$$x={\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_k...}^{-(2k)}\equiv \sum _{k=1}^{\infty }\frac{(-1{)}^k{\epsilon }_k}{2\cdot 4\cdot 8\cdot \dots \cdot 2k},\text{where}{\epsilon }_k\in \{0,1,\dots ,2k-1\};$$

$$x={\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_k\dots }^{(k+1)!}\equiv \sum _{k=1}^{\infty }\frac{{\epsilon }_k}{2\cdot 3\cdot 4\cdot \dots \cdot (k+1)},\text{where}{\epsilon }_k\in \{0,1,\dots ,k\}.$$

$$\lim \underset{k\to \infty }{\inf }\frac{|{\epsilon }_k|+1}{q_k}=0$$

$$|{\epsilon }_{k+1}|\le \frac{1}{2}\max (|{\epsilon }_k|,1)q_{k+1}.$$

$$\lim \underset{n\to \infty }{\inf }\frac{{\epsilon }_k+1}{q_k}=0$$

$${\epsilon }_{k+1}\le K\max ({\epsilon }_k,1)q_{k+1}.$$

$${\rho }_k=\{\begin{array}{ll}1& \text{if k∈NB}\\ 2& \text{if k∈/NB,}\end{array}$$

$$x={\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_k\dots }^{(\pm Q,N_B)}\equiv \frac{(-1{)}^{{\rho }_1}{\epsilon }_1}{q_1}+\frac{(-1{)}^{{\rho }_2}{\epsilon }_2}{q_1{q}_2}+\cdots +\frac{(-1{)}^{{\rho }_k}{\epsilon }_k}{q_1{q}_2\dots q_k}+\dots ,$$

$$\sigma (x)=\sigma \left({\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_k\dots }^{(\pm Q,N_B)}\right)=\sum _{k=2}^{\infty }\frac{(-1{)}^{{\rho }_k}{\epsilon }_k}{q_2{q}_3\dots q_k}=q_1{\Delta }_{0{\epsilon }_2\dots {\epsilon }_k\dots }^{(\pm Q,N_B)}.$$

$$\begin{array}{rl}{\sigma }^n(x)& ={\sigma }^n\left({\Delta }_{{\epsilon }_1{\epsilon }_2\dots {\epsilon }_k\dots }^{(\pm Q,N_B)}\right)\\ & =\sum _{k=n+1}^{\infty }\frac{(-1{)}^{{\rho }_k}{\epsilon }_k}{q_{n+1}{q}_{n+2}\dots q_k}=q_1\dots q_n{\Delta }_{\underset{n}{\underbrace{0\dots 0}}{\epsilon }_{n+1}{\epsilon }_{n+2}\dots }^{(\pm Q,N_B)}.\end{array}$$

$${\Delta }_{\underset{n}{\underbrace{0\dots 0}}{\epsilon }_{n+1}{\epsilon }_{n+2}\dots }^{(\pm Q,N_B)}=q_{n+1}\dots q_{n+m}{\Delta }_{\underset{n+m}{\underbrace{0\dots 0}}{\epsilon }_{n+m+1}{\epsilon }_{n+m+2}\dots }^{(\pm Q,N_B)}.$$

$$\lim \underset{k\to \infty }{\inf }\left(\frac{{\epsilon }_{k+1}}{q_{k+1}}-\frac{{\epsilon }_k}{q_k}\right)=0,{\epsilon }_k=o(q_{k-1}{q}_k),{\epsilon }_{k+1}-{\epsilon }_k=o(q_{k-1}{q}_k).$$

$$S_N=\sum _{k=N}^{\infty }\frac{{\epsilon }_n}{q_N\cdots q_k}.$$

$$S_{k+1}-S_k<\epsilon$$

Full text

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11K55 11J72 26A30 \VOLUME31 \YEAR2023 \NUMBER1

\DOIhttps://doi.org/10.46298/cm.10454 {paper}

1 Introduction

Let $Q\equiv(q_{k})$ be a fixed sequence of positive integers, $q_{k}>1$, $\Theta_{k}$ be a sequence of the sets $\Theta_{k}\equiv\{0,1,\dots,q_{k}-1\}$, and $\varepsilon_{k}\in\Theta_{k}$. The Cantor series expansion

[TABLE]

of $x\in[0,1]$, first studied by G. Cantor in [Cantor1], is a natural generalization of the b-ary expansion

[TABLE]

of numbers from the closed interval $[0,1]$. Here $b$ is a fixed positive integer, $b>1$, and $\alpha_{n}\in\{0,1,\dots,b-1\}$. By $x=\Delta^{Q}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{k}\ldots}$ denote a number $x\in[0,1]$ represented by series (1). This notation is called the representation of $x$ by Cantor series (1). We note that certain numbers from $[0,1]$ have two different representations by Cantor series (1), i.e.,

[TABLE]

Such numbers are called $Q$-rational. The other numbers in $[0,1]$ are called $Q$-irrational. Cantor series expansions have been intensively studied from different points of view during the last century. The metric, probability, and fractal theories of number representations by positive Cantor series were studied by a number of researchers. Also, functions and fractal sets defined in terms of Cantor series expansions were investigated. These problems were considered by the following researchers: P. Erdös, J. Galambos, G. Iommi, P. Kirschenhofer, T. Komatsu, V. Laohakosol, B. Li, M. Paštéka, S. Prugsapitak, J. Rattanamoong, A. Rényi, B. Skorulski, R. F. Tichy, P. Turán, Yi Wang, M. S. Waterman, H. Wegmann, Liu Wen, Zhixiong Wen, Lifeng Xi, and other mathematicians. Such investigations can be divided into two groups. The first is the investigation of the fractional parts of real numbers represented by Cantor series (1), and the other is the investigation of representations of non-negative integers represented by positive Cantor series of the form

[TABLE]

where $\varepsilon_{k}\in\Theta_{k}$. We give a brief description of these investigations. A number of researches are devoted to studying various types of the normality of numbers represented by the Cantor series. In these papers, the notions of Q-distribution normality, Q-normality, and Q-ratio normality, are studied. For example, in the papers [Beros2014], [Mance_2012], [Mance_2014], the notion of Q-distribution normality is investigated. Indeed, one can note the following investigations: relations between various types of normality (e.g., see [Beros2014], [Bill10]); the average value of the function of the sum of digits in the Cantor series representation of a number (see [Kirschenhofer_Tichy84] and references in the last-mentioned article); behaviour of the frequency of the most frequently used digit among the first digits in the representation of a number (e.g., see [Erdos_Renyi1959]); necessary, sufficient, necessary and sufficient conditions for a number to be a number having the property of certain type normality (see [Mance_2012], [Bill10], [Mance_2014]); the completeness of the Lebesgue measure, the density, topological properties, the Hausdorff measure of a set whose elements are numbers having the property of the normality of a certain type (e.g., see [Mance_2012], [Mance_2014]); the rationality and irrationality of a number which has the property of the normality of a certain type (see [Bill10]), etc. Note that, in the papers [Erdos_Renyi1959], [Erdos_Renyi1959(2)], [Renyi55], [Renyi56], [Renyi58], [Turan56], P. Erdös, A. Rényi, and P. Turán introduced and studied the problem on normal numbers and other statistical properties of real numbers with respect to large classes of Cantor series expansions. Some investigations of Cantor series expansions were published by J. Galambos in [Galambos1976], [Galambos1976(2)]. In some papers, certain generalizations of real numbers representations by the Cantor series are studied. For example, properties of digits (sequences of digits) of the polyadic number $\alpha$ as functions (sequences of functions) of $\alpha$ are studied in [Pasteka1996]; in [KLPR????], the notion of a complex Cantor series is introduced, and the $\mathbb{Q}$-algebraic and $\mathbb{Q}$-linearly independence of numbers represented by Cantor series are investigated; matrix expansions are studied in [Waterman1975]; the papers [Serbenyuk2016], [Serbenyuk2018] are devoted to certain generalizations of alternating Cantor series. In certain papers, fractal properties of representations of real numbers by positive Cantor series and fractal properties of certain type sets whose elements are represented by a positive Cantor series, are studied (e.g., see [Iommi_Skorulski_2009], [Wang_Wen_Xi2009], [Mance_2015], [Bill10]). For example, in [Iommi_Skorulski_2009], the Hausdorff-Besicovitch dimensions of sets whose elements are defined in terms of the frequencies of digits, are investigated. The paper [Wegmann1968] is devoted to studying the conditions under which the family of all possible rank cylinders $\Delta^{Q}_{c_{1}c_{2}\ldots c_{n}}$ is faithful for the Hausdorff-Besicovitch dimension calculation. Sets whose elements have a restriction on using digits in their own representations are studied in [Mance_2015]. In the last-mentioned article, the formula for a calculation of the Hausdorff dimension of the following set is proved, and conditions for the equality of the Hausdorff, packing, and box dimensions of this set, are discovered:

[TABLE]

Here the condition

[TABLE]

holds. Also, we can note several investigations of functions. The arguments or values of these functions are defined by positive [Cantor1] or alternating [Serbenyuk2017] Cantor series. In [LiuWen2001], properties of the following function were investigated:

[TABLE]

where $u_{1}=1$ and for $k=1,2,\dots$

[TABLE]

Here $x$ is represented by series (1). This function is well-defined and continuous. Also, $u=f(x)$ is nowhere differentiable when $q_{k}\geq 3$ for all $k=1,2,\dots,$ and the condition

[TABLE]

holds. The last-mentioned function is a function with a complicated local structure. Certain examples of functions with a complicated local structure are described in [S.Serbenyukpreprint2], [Symon2015], [Symon2017]. In the paper [Mance_2015], the following function are studied:

[TABLE]

where

[TABLE]

Here $E_{0},\varepsilon_{0}\in\mathbb{Z}$, $Q\equiv(q_{k})$ and $P\equiv(p_{k})$ are sequences of positive integers, that greater than $1$. Also, $E_{k}\neq p_{k}-1$ and $\varepsilon_{k}\neq q_{k}-1$ infinitely often, $E_{k}\in\{0,1,\dots,p_{k}-1\}$ and $\varepsilon_{k}\in\Theta_{k}$. In the present article, the main attention is given to necessary and sufficient conditions for $x$ (represented by Cantor series with an arbitrary basic sequence $(q_{k})$) to be rational.

Remark 1.1**.**

In the present article, we use the following notations: $\mathbb{N}$, $\mathbb{Z}_{0}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{I}$. Here by $\mathbb{N}$ denote the set of all positive integers and by $\mathbb{Z}_{0}$ denotes the set $\mathbb{N}\cup\{0\}$, $\mathbb{Z}$ is the set of all integers, and $\mathbb{Q}$ is the set of all rational numbers, and $\mathbb{I}$ is the set of all irrational numbers.

2 Description of research of the main problem

The problem of expansions of rational/irrational numbers in terms of generalizations of the b-ary numeral system is difficult. A version of this problem for Cantor series (1) was introduced in the paper [Cantor1] in 1869 and has been studied by a number of researchers. For example, G. Cantor, P. A. Diananda, A. Oppenheim, P. Erdös, J. Hančl, E. G. Straus, P. Rucki, R. Tijdeman, P. Kuhapatanakul, V. Laohakosol, D. Marques, Pingzhi Yuan and other scientists studied this problem. In the monograph [Galambos1976], Prof. János Galambos called the problem on representations of rational numbers by Cantor series (1) as the fourth open problem, and wrote the following: “Problem Four. Give a criterion of rationality for numbers given by a Cantor series. What one should seek here is a directly applicable criterion. A general sufficient condition for rationality would also be of interest, in which the quoted theorems of Diananda and Oppenheim (including the abstract criterion by condensations) can be guides or useful tools. If in a Cantor series, negative and positive terms are permitted, somewhat less is known about the rationality or irrationality of the resulting sum. G. Lord (personal communication) tells me that the condensation method can be extended to this case as well, but still, the results are less complete than in the case of ordinary Cantor series.”([Galambos1976, p. 134]). The paper [Serbenyuk21] is devoted to the last-mentioned discussion and to expansions of rational numbers by sign-variable Cantor series. For fullness, one can note the following result of Diananda and Oppenheim noted by J. Galambos.

Theorem 2.1** ([Diananda_Oppenheim1955]).**

A necessary and sufficient condition that $x$ given by (1) shall be rational is this: coprime integers $h,k$, $0\leq h\leq k$, an integer $N$ and a condensation shall exist such that

[TABLE]

for all $i\geq N$.

Here

[TABLE]

where $A_{0}=\varepsilon_{0}$ is the integer part of $x$,

[TABLE]

and $B_{i}\geq 2$, $0\leq A_{i}\leq B_{i}-1$,

[TABLE]

We begin with a brief description of investigations of rational numbers represented by the Cantor series. Much research [Tijdeman_Pingzhi2002], [Hancl97], [Hancl_Tijdeman2004], [Hancl2002], [Hancl_Tijdeman2004(2)] has been devoted to necessary or/and sufficient conditions for a rational number to be representable by Cantor series (1) such that sequences $(q_{k})$ and $(\varepsilon_{k})$ are sequences of integers. In some papers (see [Hancl_Tijdeman2004], [Tijdeman_Pingzhi2002],[Erdos_Straus1974], [Hancl2002], [Bill10]), the case of Cantor series for which sequences $(q_{k})$ and $(\varepsilon_{k})$ are sequences of integers and the condition $\mathbb{Z}\ni q_{k}>1$ holds for all $k\in\mathbb{N}$, is investigated. However, the main problem of the present article is studied for the case of series (1) (e.g., see [Cantor1], [Diananda_Oppenheim1955], [Kuhapatanakul_Laohakosol2001], [Oppenheim1954]) and still for the case of Cantor series of a special type (e.g., see [Hancl_Tijdeman2005], [Hancl_Tijdeman2010], [Hancl_Tijdeman2009], [Hancl_Tijdeman2004]). For example, in the papers [Diananda_Oppenheim1955], [Hancl_Tijdeman2004(2)], [Erdos_Straus1968], [KN2016], Ahmes series are considered. The last series is Cantor series (1) for which $\varepsilon_{k}=const=1$ holds for all $k\in\mathbb{N}$. In the papers [Diananda_Oppenheim1955], [Hancl97], [Hancl2002], [Hancl_Tijdeman2004], [Hancl_Tijdeman2004(2)], [Hancl_Tijdeman2005],[Erdos_Straus1974], [Oppenheim1954], [Tijdeman_Pingzhi2002], necessary and sufficient conditions for a rational (irrational) number to be representable by a Cantor series are studied, and sufficient conditions are investigated in the papers [Erdos_Straus1974], [Diananda_Oppenheim1955], [Hancl_Tijdeman2004], [Kuhapatanakul_Laohakosol2001], [Oppenheim1954], [Tijdeman_Pingzhi2002]. Although much research has been devoted to the problem of representations of rational (irrational) numbers by Cantor series for which sequences $(q_{k})$ and $(\varepsilon_{k})$ are sequences of special types (see [Cantor1],[Erdos_Straus1974], [Hancl97], [Hancl2002], [Hancl_Tijdeman2004], [Kuhapatanakul_Laohakosol2001], [Oppenheim1954],[Tijdeman_Pingzhi2002]), little is known about necessary and sufficient conditions of the rationality (irrationality) for the case of an arbitrary sequence $(q_{k})$ (see [Diananda_Oppenheim1955], [Hancl_Tijdeman2004], [Serbenyuk:Cantorseries], [S13], [Serbenyuk2017], [Rationalnumbers2018], [Serbenyuk21], [Tijdeman_Pingzhi2002]). Finally, several papers (see [Hancl_Rucki2006], [KN2016], [Kuhapatanakul_Laohakosol2001], [Tijdeman_Pingzhi2002]) were devoted to investigations of conditions of the rationality or irrationality of numbers represented by series of the form $\sum^{\infty}_{k=1}{\frac{a_{k}}{b_{k}}}$. Furthermore, in [Kuhapatanakul_Laohakosol2001], a necessary and sufficient condition of the rationality of the sum $\sum^{\infty}_{k=1}{\frac{a_{k}(-1)^{k+1}}{b_{k}}}$ is proved for the case of certain properties which are satisfied by sequences $(a_{k})$ and $(b_{k})$. Let us consider our problem more in detail.

3 Cantor’s investigations, finite expansions, and conditions for finite expansions of rational numbers

Let us begin with a consideration of the results presented in the first paper on this topic (i.e., [Cantor1]). In [Cantor1], G. Cantor proved a fact that an arbitrary number $x\in[0,1)$ is a rational number if and only if $(\varepsilon_{k})$ is ultimately periodic under the condition when a sequence $(q_{k})$ is periodic. In addition, one can note the following theorem which necessity was given in [Cantor1] with the other formulation and with a more complicated proof for the case of positive Cantor series.

Theorem 3.1**.**

A rational number $\frac{p}{r}$ has a finite expansion by a positive or sign-variable Cantor series if and only if there exists a number $n_{0}$ such that

[TABLE]

The interest in the last theorem can be explained ([Serbenyuk:Cantorseries], [S13], [Serbenyuk2017], [Serbenyuk21]) by the fact that there exist certain sequences $(q_{k})$ such that all rational numbers represented by Cantor series (positive or sign-variable) have finite expansions. For example, all rational numbers represented by the following representations have finite expansions.

[TABLE]

[TABLE]

It is easy to see that there exist sequences $(q_{k})$ and $(\varepsilon_{k})$ such that a finite expansion is a necessary or/and sufficient condition of the rationality of any number represented by a Cantor series. Several papers were devoted to such investigations. For example, see [Kuhapatanakul_Laohakosol2001], [Hancl2002]. Let us consider several related results. In 2006, J. Sondow gave a geometric proof of the irrationality of the number $e$ [Sondow2006]. In [Marques2009], the following statement was proved by a generalization to Sondow’s construction.

Theorem 3.2** ([Marques2009]).**

Let $x=\sum^{\infty}_{k=1}{\frac{\varepsilon_{n}}{q_{1}q_{2}\cdots q_{k}}}$. Suppose that each prime divides infinitely many of the $q_{k}$. Then $x\in\mathbb{I}$ if and only if both $0<\varepsilon_{k}<q_{k}-1$ hold infinitely often.

For example, in [Hancl2002], attention is given to conditions of finite expansions of rational numbers by positive and sign-variable Cantor expansions. That is, $\sum^{\infty}_{n=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if $\varepsilon_{k}=0$ for every sufficiently large positive integer $k$ under one of the following two systems of conditions:

• •

System 1 of conditions (the case of sign-variable series): suppose $(q_{k})$ is a sequence of positive integers greater than one, $(\varepsilon_{k})$ is a sequence of integers such that the condition

[TABLE]

holds and for every sufficiently large positive integer $k$

[TABLE]

• •

System 2 of conditions (the case of positive series): suppose $(q_{k})$ is a sequence of positive integers greater than one and $K\in(0,1)$, $(\varepsilon_{k})$ is a sequence of non-negative integers such that the condition

[TABLE]

holds and for every sufficiently large positive integer $k$

[TABLE]

4 The shift operator and related investigations

We must note that the notion of the shift operator plays an important role in investigations of expansions of rational numbers defined by the Cantor series (positive, alternating, or sign-variable). We begin with definitions. Let $N_{B}$ be a fixed subset of positive integers,

[TABLE]

and $Q\equiv(q_{k})$ be a fixed sequence of positive integers such that $q_{k}>1$ for all $n\in\mathbb{N}$. Then we get the following representation of real numbers

[TABLE]

where $\varepsilon_{k}\in\{0,1,\dots,q_{k}-1\}$. The last representation is called the representation of a number $x$ by a sign-variable Cantor series or the quasi-nega-Q-representation. It is easy to see that we get a positive Cantor series whenever $N_{B}=\emptyset$. Define the shift operator $\sigma$ of expansion (2) by the rule

[TABLE]

Clearly,

[TABLE]

The following theorem is the most general statement on the representation of rational numbers for any sequences $(q_{k})$, $(\varepsilon_{k})$, and an arbitrary set $N_{B}$.

Theorem 4.1** ([Serbenyuk:Cantorseries], [S13], [Serbenyuk2017], [Rationalnumbers2018]).**

A number $x$ represented by series (2) is rational for the case of any $N_{B}\subseteq~{}\mathbb{N}$ if and only if there exist numbers $n\in\mathbb{Z}_{0}$ and $m\in\mathbb{N}$ such that $\sigma^{n}(x)=\sigma^{n+m}(x)$.

The last theorem can be formulated by the following way.

Theorem 4.2** ([Serbenyuk:Cantorseries], [S13], [Rationalnumbers2018]).**

A number $x=\Delta^{(\pm Q,N_{B})}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{k}\ldots}$ is rational if and only if there exist numbers $n\in\mathbb{Z}_{0}$ and $m\in\mathbb{N}$ such that

[TABLE]

Let us recall several auxiliary statements which are true for positive Cantor series but do not hold for the general case of sign-variable Cantor series (i.e., for certain sets $N_{B}$).

Lemma 4.3** ([Serbenyuk:Cantorseries], [S13]).**

Let $n_{0}$ be a fixed positive integer. Then the condition $\sigma^{n}(x)=const$ holds for all $n\geq n_{0}$ if and only if $\frac{\varepsilon_{n}}{q_{n}-1}=const$ for all $n>n_{0}$.

Lemma 4.4** ([Serbenyuk:Cantorseries], [S13]).**

Suppose we have $q=\min_{n\in\mathbb{N}}{q_{n}}$ and fixed $\varepsilon\in\{0,1,\dots,q-1\}$. Then the condition $\sigma^{n}(x)=x=\frac{\varepsilon}{q-1}$ holds if and only if the condition $\frac{q_{n}-1}{q-1}\varepsilon=\varepsilon_{n}\in\mathbb{Z}_{0}$ holds for all $n\in\mathbb{N}$.

Let us consider cases when the condition $\frac{\varepsilon_{k}}{q_{k}-1}=const$ (the last equality holds for all $k$ greater than some fixed $k_{0}$) is a necessary and/or sufficient condition for a rational number to be representable by a positive Cantor series. For more information, see [Diananda_Oppenheim1955], [Hancl_Tijdeman2004], [Tijdeman_Pingzhi2002]. In [Hancl_Tijdeman2004], J. Hančl and R. Tijdeman formulated certain conditions of the irrationality of a number represented by Cantor series (1) when sequences $(q_{k})$ and $(\varepsilon_{k})$ are sequences of positive integers and $q_{k}>1$ for all $k\in\mathbb{N}$. Applications of the shift operator to representations of rational numbers by such series are considered. This article is partially devoted to conditions under which the condition $\frac{\varepsilon_{k}}{q_{k}-1}=const$ is a necessary and sufficient condition of the rationality of numbers represented by such expansions. In particular, the following cases are considered:

[TABLE]

Also, in [Hancl_Tijdeman2004], the authors noted that sum (1) is equal to a rational number if $\frac{\varepsilon_{k}}{q_{k}-1}=const$ holds for all $k$ greater than some number $n_{0}$. Let us recall some results.

Lemma 4.5** ([Hancl_Tijdeman2004]).**

If $S=\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\ldots q_{k}}}=\frac{r}{p}$ holds for a certain $r\in\mathbb{Z}$ and $p\in\mathbb{N}$, then $pS_{N}\in\mathbb{Z}$ for all $N\in\mathbb{N}$.

Here $S=\sigma^{0}(x)$ and $S_{N}=\sigma^{N-1}(x)$. That is,

[TABLE]

Proposition 4.6** ([Hancl_Tijdeman2004]).**

If $(S_{k})$ is bounded from below and for every $\varepsilon>0$ we have

[TABLE]

for $k\geq k_{0}(\varepsilon)$, then $S=\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{k}}{q_{k}-1}=const$ for $N>N_{0}$.

Corollary 4.7** ([Hancl_Tijdeman2004]).**

If $(\varepsilon_{k})$ is a sequence of positive integers such that $\varepsilon_{k+1}-\varepsilon_{k}=o(k)$, then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{k!}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{k}}{k-1}=const$ for $k$ greater than some $k_{1}$.

Theorem 4.8** ([Hancl_Tijdeman2004]).**

Let $(q_{k})$ be a sequence of positive integers which is monotonic and satisfies $\varepsilon_{k}=o(q^{2}_{k})$. Then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{k}}{q_{k}-1}=const$ for $k\geq k_{0}$.

Theorem 4.9** ([Hancl_Tijdeman2004]).**

Let $(q_{k})$ and $(\varepsilon_{k})$ be sequences of integers such that $q_{k}>1$ for all $k\in\mathbb{N}$. If $\left(\frac{\varepsilon_{k}}{q_{k}}\right)$ is bounded from below, $\lim_{k\to\infty}{\frac{\varepsilon_{k}}{q_{k-1}q_{k}}}=0$, and for each $\varepsilon>0$ there exists $k_{0}(\varepsilon)$ such that the condition $\frac{\varepsilon_{k+1}}{q_{k+1}}<\frac{\varepsilon_{k}}{q_{k}}+\varepsilon$ holds for $k>k_{0}(\varepsilon)$, then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{k}}{q_{k}-1}=const$ for $k\geq N_{0}$.

Theorem 4.10** ([Hancl_Tijdeman2004]).**

Let $(q_{k})$ be a monotonic sequence of positive integers satisfying $\lim_{k\to\infty}{\frac{q_{k}}{\log k}}=\infty$. Then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{k}}{q_{k}-1}=const$ for $k\geq k_{0}$.

Theorem 4.11** ([Hancl_Tijdeman2004]).**

Let $(q_{k})$ be an unbounded monotonic sequence of positive integers. Then $\sum^{\infty}_{k=1}{\frac{k}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if $\frac{k}{q_{k}-1}=const$ for $k\geq k_{0}$.

Results obtained in [Hancl_Tijdeman2004] were generalized and corrected by Robert Tijdeman and Pingzhi Yuan in paper [Tijdeman_Pingzhi2002]. In particular, results are generalized for the cases when $\varepsilon_{k}=k$ and $q_{k}\to\infty$, $q_{k}=k$ and $\varepsilon_{k+1}-\varepsilon_{k}=O(k)$. In the last-mentioned article, it is shown that, in order that the condition $\frac{\varepsilon_{k}}{q_{k}-1}=const$ for all $k\geq k_{0}$ is a necessary and sufficient condition of the rationality, one can neglect the condition $\varepsilon_{k}=o(q^{2}_{k})$ in the system of conditions: $\varepsilon_{k}=o(q^{2}_{k})$ , $\varepsilon_{k}\geq 0$, $\varepsilon_{k+1}-\varepsilon_{k}<\varepsilon q_{k}$ for $k\geq k_{1}(\varepsilon)$. We note the following statements.

Theorem 4.12** ([Tijdeman_Pingzhi2002]).**

Let $(q_{n})$ be a monotonic integer sequence with $q_{n}>1$ for all $n$ and $(\varepsilon_{n})$ be an integer sequence such that $\varepsilon_{n+1}-\varepsilon_{n}=o(q_{n+1})$. Then $\sum^{\infty}_{n=1}{\frac{\varepsilon_{n}}{q_{1}q_{2}\cdots q_{n}}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{n}}{q_{n}-1}=const$ for all $n$ greater than some $n_{0}$.

Theorem 4.13** ([Tijdeman_Pingzhi2002]).**

Let $(q_{k})$ be a monotonic sequence of positive integers, $q_{k}>1$. Let $(\varepsilon_{k})$ be a sequence of positive integers satisfying

[TABLE]

Then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\ldots q_{k}}}\in\mathbb{Q}$ if and only if $\frac{\varepsilon_{k}}{q_{k}-1}=const$ for all $k$ greater than some $k_{0}$.

In addition, the following sufficient condition of the irrationality is proved.

Theorem 4.14** ([Tijdeman_Pingzhi2002]).**

Let $q_{k}>1$ be such that $\varepsilon_{k}=O(q_{k})$ for all $k$ and $\lim_{k\to\infty}{\frac{\varepsilon_{k}}{q_{k}}}=\alpha\in\mathbb{I}$. Then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{I}$.

The last statement with the condition $0\leq\varepsilon_{k}<q_{k}$ without $\varepsilon_{k}=O(q_{k})$ was proved in [Oppenheim1954]. Finally, in [Tijdeman_Pingzhi2002], the following denotations are used in proofs:

[TABLE]

Here $(n_{k})$ is a subsequence of positive integers, $n_{0}=1$,

[TABLE]

and $q^{*}_{k}=q_{n_{k}-1}q_{n_{k}-2}\cdots q_{n_{k-1}}$, $k=1,2,3,\ldots$. For series (1), where $(q_{n})$ and $(\varepsilon_{n})$ are sequences of integers such that $q_{n}>0$ for all $n\in\mathbb{N}$ and series (1) converges, the following statements are true.

Lemma 4.15** ([Tijdeman_Pingzhi2002]).**

Using the notation above, if there exists a subsequence $(n_{k})$ of positive integers such that $R_{n_{k}}=R_{n_{k+1}}$ for $k=1,2,\dots$, then $S\in\mathbb{Q}$.

Proposition 4.16** ([Tijdeman_Pingzhi2002]).**

If $(R_{n})$ is bounded from below and there exists a subsequence $(n_{k})$ of positive integers with $R_{n_{k+1}}-R_{n_{k}}<\varepsilon$ for $k\geq k_{0}(\varepsilon)$, then $S\in\mathbb{Q}$ if and only if $R_{n_{k}}=R_{n_{k+1}}$ for all large $k$.

In [Oppenheim1954], A. Oppenheim studied sufficient conditions of the irrationality of numbers represented by Cantor series (1) and, also, alternating series (1) such that $|\varepsilon_{i}|<q_{i}-1$ for $i=1,2,3,\dots$, and $\varepsilon_{m}\varepsilon_{n}<0$ for some $m>i$ and $n>i$ when $i$ is any fixed integer. Also, in [Oppenheim1954], the main results obtained by using some results from [Cantor1] and sums of the form

[TABLE]

where $(i_{k})$ is some subsequence of positive integers, and by investigation of the limit of $c_{i_{k}}=\frac{\varepsilon_{i_{k}}}{q_{i_{k}}}$ as $k\to\infty$. That is, here $x_{i_{k}}=\sigma^{i_{k}-1}(x)$.

Lemma 4.17** ([Oppenheim1954]).**

A necessary and sufficient condition that $x$ given by convergent series (1), where $q_{k}$ and $\varepsilon_{k}$ are integers, shall be irrational is that for every integer $p\in\mathbb{N}$ we can find an integer $r\in\mathbb{Z}$ and a subsequence $(i_{k})$ such that

[TABLE]

Finally, in this section, we note necessary and sufficient conditions for a rational number to be representable by certain types of Cantor series which were investigated by P. Erdös and E. G. Straus in [Erdos_Straus1974].

Theorem 4.18** ([Erdos_Straus1974]).**

Let $(\varepsilon_{n})$ be a sequence of integers and $(q_{k})$ be a sequence of positive integers with $q_{k}>1$ for all large $k$ and

[TABLE]

Then $\sum^{\infty}_{k=1}{\frac{\varepsilon_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{Q}$ if and only if there exist a positive integer $B$ and a sequence of integers $(c_{k})$ such that for all large $k$ we have

[TABLE]

Theorem 4.19** ([Erdos_Straus1974]).**

Let $p_{k}$ be the $k$th prime and let $(q_{k})$ be a monotonic sequence of positive integers satisfying

[TABLE]

Then $\sum^{\infty}_{k=1}{\frac{p_{k}}{q_{1}q_{2}\cdots q_{k}}}\in\mathbb{I}$.

5 Certain approaches to investigations of expansions of rational numbers

We considered mainly the shift operator. Now one can consider another approaches to investigations of expansions of rational numbers but some of them are related with the shift operator. In [Bill10], the probabilistic approach is used and the attention is given to Cantor series (1) for which $\varepsilon_{k}\neq q_{k}-1$ infinitely often. Irrationality of numbers having a property of a certain type of the normality is investigated.

Definition 5.1** ([Bill10, p.45]).**

A number $x\in[0,1)$ is called Q-distribution normal if the sequence

[TABLE]

is uniformly distributed in $[0,1)$.

Theorem 5.2** ([Bill10, p. 264]).**

A number $x\in[0,1)$ is irrational if and only if there exists a basic sequence $Q=(q_{k})$ such that $x$ is Q-distribution normal.

In the paper [Kumar2019], the subspace theorem is used for proving conditions for a transcendental number to be representable by positive Cantor series. Such conditions were formulated in terms of blocks of digits $\varepsilon_{k}$ and in terms of tuples of digits for expansion (1). Finally, one approach based on the notion of cylinders of Cantor expansions gives an opportunity to model rational numbers. However, we have necessary and sufficient conditions for the case of the positive Cantor series and a necessary condition for the sign-variable Cantor series. Let us consider the following two theorems.

Theorem 5.3** ([Rationalnumbers2018]).**

A number $x=\Delta^{Q}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}\ldots}\in(0,1)$ represented by series (1) is a rational number $\frac{p}{r}$, where $p,r\in\mathbb{N},(p,r)=1$, and $p<r$, if and only if the condition

[TABLE]

holds for all $1<n\in\mathbb{N}$, where $\Delta_{1}=pq_{1}$, $\varepsilon_{1}=\left[\frac{\Delta_{1}}{r}\right]$, and $[a]$ is the integer part of $a$.

For fullness, we give some examples of rational numbers from [Rationalnumbers2018]. Really, suppose

[TABLE]

Then

[TABLE]

Theorem 5.4** ([Serbenyuk21]).**

If $x=\Delta^{(\pm Q,N_{B})}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{k}\ldots}=\frac{p}{r}$, where $p\in\mathbb{Z},r\in\mathbb{N},(|p|,|r|)=1$, and $|p|<r$, then the condition

[TABLE]

holds for all $1<k\in\mathbb{N}$. Here $\Delta^{(g)}_{1}=pq_{1}$, $\varepsilon_{1}=\left|\left[\frac{\Delta^{(g)}_{1}}{r}+s_{1}\right]\right|$, and $[a]$ is the integer part of $a$. Also,

[TABLE]

One can note that the two last statements are related to the shift operator. Really, in the case of positive Cantor series [Rationalnumbers2018], we have $\sigma^{n}(x)=\{\frac{\Delta_{n}}{r}\}$ and $\varepsilon_{n}=[\frac{\Delta_{n}}{r}]$. In the general case of sign-variable Cantor series (i.e., there is no number $k_{0}$ such that any $k\in N_{B}$ or any $k\notin N_{B}$ for all $k>k_{0}$), we obtain [Serbenyuk21] the following:

[TABLE]

where $\{a\}$ is the fractional part of $a$ (i.e., $a=[a]+\{a\}$). In this survey, we have demonstrated the main conditions for a rational number to be representable by positive, alternating, and sign-variable Cantor series. Connections among some of them are described. An important role of the notion of the shift operator in investigations in this topic, is noted.