Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet
Mykola Pratsiovytyi, Oleh Vynnyshyn
TL;DR
This study analyzes numeral systems with base $s$ and a redundant alphabet $A_r=\{0,\dots,r\}$ for $s\le r\le 2s-2$, focusing on numbers in $\left[0,\dfrac{r}{s-1}\right]$ with $r_s$-representations. It derives a necessary-and-sufficient uniqueness criterion, shows that the unique-representation set has Hausdorff--Besicovitch dimension $\dfrac{\ln(2s-r-1)}{\ln s}$, and characterizes purely periodic representations, revealing a Cantor-type structure and a symmetry about $\dfrac{r}{2(s-1)}$. The results establish that almost all numbers have a continuum of representations (full Lebesgue measure), while numbers with finitely or countably many representations form a measure-zero set, highlighting a sharp dichotomy between unique and continuum representations in redundant numeral systems. Collectively, the work blends fractal geometry, symbolic dynamics, and number representations to illuminate metric and topological properties of expansions in redundant bases.
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 $x=\sum\limits_{n=1}^{\infty}\frac{α_n} {s^n}\equivΔ^{r_s}_{α_1α_2...α_n...}$, $α_n\in A_r$. 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.