Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet

TL;DR

This paper explores the properties of number representations in a numeral system with a natural base and a redundant alphabet, analyzing uniqueness, fractal dimensions, and the measure of numbers with multiple representations.

Contribution

It establishes criteria for uniqueness of representations, calculates the Hausdorff dimension of the set of unique representations, and characterizes numbers with multiple representations in such systems.

Findings

Hausdorff--Besicovitch dimension of unique representation set is rac{\u2212ln(2s-r-1)}{rac{ ln s}

Set of numbers with multiple representations has full Lebesgue measure

Conditions for numbers to have continuum of representations are provided

Abstract

In this work, we study a numeral system with a natural base $s \geq 2$ and a redundant alphabet $A_r=\{0,1, \dots, r\}$, where $s \leq r \leq 2s-2$. We investigate the topological, metric, and fractal properties of the set of numbers in the interval $\left[0,\frac{r}{s-1}\right]$ that admit a unique representation $x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n} {s^n}\equiv\Delta^{r_s}_{\alpha_1\alpha_2...\alpha_n...}$, $\alpha_n\in A_r$. The criterion for the uniqueness of the number representation is established. It is proved that the Hausdorff--Besicovitch dimension of the set of numbers with a unique representation is equal to $\frac{\ln(2s-r-1)}{\ln s}$. An analysis of the quantity of representations of numbers having purely periodic representations with a simple period (a single-digit period) is carried out. It is proved that the set of numbers that admit a continuum of distinct representations has full Lebesgue measure. Conditions for a number to belong to this set are given in terms of one of its representations.