Modeling rational numbers by Cantor series

TL;DR

This paper explores how rational numbers can be modeled using Cantor series, establishing relations between their digits in these representations for arbitrary sequences.

Contribution

It provides new proofs of relations between digits in Cantor series representations of rational numbers with arbitrary sequences.

Findings

Relations between digits are established for rational numbers in Cantor series.

Proofs are provided for arbitrary sequences $(q_k)$ in Cantor series.

The paper enhances understanding of rational number representations via Cantor series.

Abstract

In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case of an arbitrary sequence $(q_k)$) are proved.

Equations

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

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

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

$${\Delta }_{c_1{c}_2...c_m}^Q\equiv \{x:x={\Delta }_{c_1{c}_2...c_m{\epsilon }_{m+1}{\epsilon }_{m+2}\dots {\epsilon }_{m+k}\dots }^Q\}.$$

$$\left[{\Delta }_{c_1{c}_2...c_{m}000}^Q,{\Delta }_{c_1{c}_2...c_m[q_{m+1}][q_{m+2}][q_{m+3}]...}^Q\right].$$

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

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

$$x=\sum _{i=1}^n\frac{{\epsilon }_i}{q_1{q}_2\dots q_i}+\frac{1}{q_1{q}_2\dots q_n}{\sigma }^n(x).$$

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

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

$$\frac{1}{w}=\sum _{k=1}^{\infty }\frac{{\epsilon }_k^{{}^{\prime }}}{q_1{q}_2\cdots q_k},$$

$$\frac{{\lambda }_k}{{\mu }_k}=\frac{{\epsilon }_{n_k+1}{q}_{n_k+2}\dots q_{n_{k+1}}+{\epsilon }_{n_k+2}{q}_{n_k+3}\dots q_{n_{k+1}}+\cdots +{\epsilon }_{n_{k+1}-1}{q}_{n_{k+1}}+{\epsilon }_{n_{k+1}}}{q_{n_k+1}{q}_{n_k+2}\dots q_{n_{k+1}}-1}=const;$$

$$x=\sum _{k=1}^{\infty }\frac{{\epsilon }_k}{q_1{q}_2\dots q_k}=\sum _{j=1}^{n_1}\frac{{\epsilon }_j}{q_1{q}_2\dots q_j}+\frac{1}{q_1{q}_2\dots q_{n_1}}{x}^{{}^{\prime }},$$

$$x^{{}^{\prime }}=\sum _{k=1}^{\infty }\frac{{\epsilon }_{n_k+1}{q}_{n_k+2}{q}_{n_k+3}\dots q_{n_{k+1}}+{\epsilon }_{n_k+2}{q}_{n_k+3}\dots q_{n_{k+1}}+\cdots +{\epsilon }_{n_{k+1}-1}{q}_{n_{k+1}}+{\epsilon }_{n_{k+1}}}{(q_{n_1+1}\dots q_{n_2})(q_{n_2+1}\dots q_{n_3})\dots (q_{n_k+1}\dots q_{n_{k+1}})}$$

$$=\sum _{k=1}^{\infty }\frac{{\lambda }_k}{({\mu }_1+1)\dots ({\mu }_k+1)}.$$

$$x={\Delta }_{\mathrm{123...}k...}^{(2k+1)}\equiv \frac{1}{3}+\frac{2}{3\cdot 5}+\frac{3}{3\cdot 5\cdot 7}+\cdots +\frac{k}{3\cdot 5\cdot \dots \cdot (2k+1)}+\cdots =\frac{1}{2}.$$

$$\frac{1}{6}={\Delta }_{\mathrm{0234...}k...}^{(2k+1)},\frac{5}{6}={\Delta }_{\mathrm{2234...}k...}^{(2k+1)}.$$

$$\frac{p}{r}=\sum _{n=1}^{\infty }\frac{{\delta }_n}{q_1{q}_2\cdots q_n}.$$

$${\Delta }_{c_1{c}_2...c_{m-1}[c_m]\mathrm{000...}}^Q\le x<{\Delta }_{c_1{c}_2...c_{m-1}[c_m+1]\mathrm{000...}}^Q.$$

$$\frac{p}{r}\in {\Delta }_{{\epsilon }_1}^Q=\left[{\Delta }_{{\epsilon }_1\mathrm{000...}}^Q,{\Delta }_{{\epsilon }_1[q_2-1][q_3-1]...}^Q\right]=\left[\frac{{\epsilon }_1}{q_1},\frac{{\epsilon }_1+1}{q_1}\right].$$

$$\frac{{\epsilon }_1}{q_1}\le \frac{p}{r}<\frac{{\epsilon }_1+1}{q_1},$$

$${\epsilon }_{1}r\le p{q}_1<r({\epsilon }_1+1),$$

$${\epsilon }_1\le \frac{p{q}_1}{r}<{\epsilon }_1+1.$$

$${\epsilon }_1=\left[\frac{p}{r}{q}_1\right]\equiv {\delta }_1,$$

$$\frac{p}{r}\in {\Delta }_{{\delta }_1{\epsilon }_2}^Q=\left[\frac{q_2{\delta }_1+{\epsilon }_2}{q_1{q}_2},\frac{q_2{\delta }_1+{\epsilon }_2+1}{q_1{q}_2}\right].$$

$$\frac{q_2{\delta }_1+{\epsilon }_2}{q_1{q}_2}\le \frac{p}{r}<\frac{q_2{\delta }_1+{\epsilon }_2+1}{q_1{q}_2},$$

$${\epsilon }_2\le \frac{p{q}_1{q}_2-r{q}_2{\delta }_1}{r}<{\epsilon }_2+1.$$

$${\epsilon }_2=\left[\frac{p{q}_1{q}_2-r{q}_2{\delta }_1}{r}\right]\equiv {\delta }_2.$$

$$\frac{p}{r}\in {\Delta }_{{\delta }_1{\delta }_2{\epsilon }_3}^Q=\left[\frac{{\delta }_1{q}_2{q}_3+{\delta }_2{q}_3+{\epsilon }_3}{q_1{q}_2{q}_3},\frac{{\delta }_1{q}_2{q}_3+{\delta }_2{q}_3+{\epsilon }_3+1}{q_1{q}_2{q}_3}\right]$$

$${\epsilon }_3=\left[\frac{p{q}_1{q}_2{q}_3-r({\delta }_1{q}_2{q}_3+{\delta }_2{q}_3)}{r}\right]\equiv {\delta }_3.$$

Full text

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 [1]. It is easy to see that the Cantor series expansion is the b-ary expansion

[TABLE]

of numbers from the closed interval $[0,1]$ whenever the condition $q_{k}=b$ holds for all positive integers $k$. 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.

Let $c_{1},c_{2},\dots,c_{m}$ be an ordered tuple of integers such that $c_{i}\in\{0,1,\dots,q_{i}-~{}1\}$ for $i=\overline{1,m}$.

A cylinder $\Delta^{Q}_{c_{1}c_{2}...c_{m}}$ of rank $m$ with base $c_{1}c_{2}\ldots c_{m}$ is a set of the form

[TABLE]

That is any cylinder $\Delta^{Q}_{c_{1}c_{2}...c_{m}}$ is a closed interval of the form

[TABLE]

Define the shift operator $\sigma$ of expansion (1) by the rule

[TABLE]

It is easy to see that

[TABLE]

Therefore,

[TABLE]

Note that, in the paper [2], the notion of the shift operator of an alternating Cantor series is studied in detail.

In 1864, G. Cantor introduced the problem on representaions of rational numbers by series (1) (see [1]). More information about this problem were described in [4].

In the present article, we consider the case of positive Cantor series. In the next articles, the cases of alternating and sign-variable Cantor series and certain applications of used techniques will be consider by the author of the present article.

2. Representations of certain rational numbers

Using the main statements from [1, 3, 5] (the paper [3] is [5] translated into English), we get the following.

Proposition 1**.**

A rational number $x=\frac{p}{r}$, where $p<r$ and (p,r)=1, has a finite expansion by positive Cantor series whenever there exists a number $k_{0}$ such that the condition $q_{1}q_{2}\cdots q_{k_{0}}\equiv 0\pmod{r}$ holds.

Proposition 2**.**

There exist certain sequences $(q_{k})$ such that all rational numbers represented in terms of corresponding Cantor series have finite expansions.

For example, these representations are following:

[TABLE]

[TABLE]

Proposition 3**.**

Suppose that the sequence $(q_{k})$ is periodic. Then a number $x$ is rational if and only if the representation $\Delta^{Q}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{k}\ldots}$ of $x$ is periodic (i.e., the sequence $(\varepsilon_{k})$ is periodic).

Proposition 4**.**

The following is true:

[TABLE]

where $w\varepsilon^{{}^{\prime}}_{k}=q_{k}-1$ holds for all positive integers $k$, $w$ is a certain positive integer.

It follows from the condition $1=\Delta^{Q}_{[q_{1}-1][q_{2}-1]...[q_{k}-1]...}$.

Proposition 5**.**

Let $n_{0}$ be a fixed positive integer number, $q_{0}=\min_{n>n_{0}}{q_{n}}$, and $\varepsilon_{0}$ be a numerator of the fraction $\frac{\varepsilon_{n_{0}+k}}{q_{1}q_{2}...q_{n_{0}}q_{n_{0}+1}...q_{n_{0}+k}}$ in expansion (1) of $x$ providing that $q_{n_{0}+k}=q_{0}$. Then $\sigma^{n}(x)=const$ for all $n\geq n_{0}$ if and only if the condition $\frac{q_{n}-1}{q_{0}-1}\varepsilon_{0}=\varepsilon_{n}\in\mathbb{Z}_{0}$ holds for any $n>n_{0}$.

Proposition 6**.**

A number $x$ represented by expansion (1) is rational if and only if there exists a subsequence $(n_{k})$ of positive integers such that for all $k=1,2,\dots,$ the following conditions are true:

• •

[TABLE]

• •

$\lambda_{k}=\frac{\mu_{k}}{\mu}\lambda$, where $\mu=\min_{k\in\mathbb{N}}{\mu_{k}}$ and $\lambda$ is a number in the numerator of the fraction whose denominator equals $(\mu_{1}+1)(\mu_{2}+~{}1)\dots(\mu+~{}1)$ from sum (3).

Here

[TABLE]

[TABLE]

[TABLE]

For example,

[TABLE]

Here $x=\sigma^{n}(x)=\frac{1}{2}$ for all $n$, where $n=~{}0,1,2,\dots$.

The last-mentioned sum is useful for modeling rational numbers of the type $\frac{A}{B}$, where $B\equiv 0\pmod{2}$. In particular,

[TABLE]

3. The main results

Let $\frac{p}{r}$ be a fixed number, where $(p,r)=1$, $p<r$, and $p\in\mathbb{N},r\in\mathbb{N}$. Here $\mathbb{N}$ is the set of all positive integers. Then

[TABLE]

Remark*.*

We use the denotation $\delta_{n}$ for the known $n$th digit in the representation of a number $x$ by a Cantor series and $\varepsilon_{n}$ for the non-known $n$th digit.

In addition, since $x\in\Delta^{Q}_{c_{1}c_{2}...c_{m}}$ but $\Delta^{Q}_{c_{1}c_{2}...c_{m}[q_{m+1}-1][q_{m+2}]...}=\Delta^{Q}_{c_{1}c_{2}...c_{m-1}[c_{m}+1]000...}$, we assume that

[TABLE]

It is easy to see that

[TABLE]

That is

[TABLE]

[TABLE]

[TABLE]

So,

[TABLE]

where $[x]$ is the integer part of $x$.

Now we get

[TABLE]

Whence,

[TABLE]

[TABLE]

So,

[TABLE]

In the third step, we have

[TABLE]

and

[TABLE]

In the $n$th step, we obtain

[TABLE]

[TABLE]

Let $\varsigma_{n}$ denote the sum $\delta_{1}q_{2}q_{3}\cdots q_{n}+\delta_{2}q_{3}q_{4}\cdots q_{n}+\dots+\delta_{n-1}q_{n}$. Then

[TABLE]

[TABLE]

Denoting by $\Delta_{n}=pq_{1}q_{2}\cdots q_{n}-r\varsigma_{n}$, we get

[TABLE]

So, the following statement is true.

Lemma 1**.**

Let $x\in(0,1)$ be a rational number represented by series (1). If $x=\frac{p}{q}=\Delta^{Q}_{\delta_{1}\delta_{2}...\delta_{n}...}$, then the equality

[TABLE]

holds for all $n\in\mathbb{N}$, where

[TABLE]

Also, for $n\geq 2$ the condition $\varsigma_{n}=\varsigma_{n-1}q_{n}+\delta_{n-1}q_{n}$ holds and

[TABLE]

Lemma 2**.**

Let $x\in(0,1)$ be a rational number represented by series (1). If $x=\frac{p}{q}=\Delta^{Q}_{\delta_{1}\delta_{2}...\delta_{n}...}$, then the equality

[TABLE]

holds for all $1<n\in\mathbb{N}$, where $\Delta_{1}=pq_{1}$ and $\delta_{1}=\left[\frac{\Delta_{1}}{r}\right]$.

Suppose that the following sequence of conditions is true:

[TABLE]

[TABLE]

It follows from equality (2) that

[TABLE]

[TABLE]

and

[TABLE]

From the last-mentioned relationship and relationship (4) it follows that

[TABLE]

That is

[TABLE]

where $\{a\}$ is the fractional part of $a$.

Remark*.*

Clearly,

[TABLE]

for an arbitrary $x\in[0,1]$. However for any Q-rational number $x=\Delta^{Q}_{\delta_{1}\delta_{2}...\delta_{n-1}\delta_{n}000...}=\Delta^{Q}_{\delta_{1}\delta_{2}...\delta_{n-1}[\delta_{n}-1][q_{n+1}-1][q_{n+2}-1][q_{n+3}-1]...}$ the following conditions hold:

[TABLE]

[TABLE]

Since the condition $\sigma^{n}\left(x\right)=1$ holds only for the last-mentioned representations of Q-rational numbers $x$, we can use only the first representation of Q-rational numbers $x$ and for these numbers the condition $\sigma^{n}\left(x\right)=0$ holds.

In addition, note that

[TABLE]

Whence for an arbitrary $n\in\mathbb{N}$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So, we have the following statement.

Theorem**.**

A number $x=\Delta^{Q}_{\delta_{1}\delta_{2}...\delta_{n}...}\in(0,1)$ 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 $\Delta_{1}=pq_{1}$, $\delta_{1}=\left[\frac{\Delta_{1}}{r}\right]$, and $[a]$ is the integer part of $a$.

Let us consider certain examples. Suppose

[TABLE]

This representation is complicated for modeling rational numbers with an even denominator:

[TABLE]

[TABLE]

Investigations of the author of the present article about representations of rational numbers by Cantor series can be useful for solving “P vs NP Problem” (this problem described in http://www.claymath.org/millennium-problems/p-vs-np-problem, www.claymath.org/sites/default/files/pvsnp.pdf). The next articles of the author of the present article will be devoted to such investigations.