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

In this work, we study a numeral system with a natural base $s\ge 2$ and a redundant alphabet $A_r=\{0,1,\dots ,r\}$, where $s\le r\le 2s-2$. We investigate the topological, metric, and fractal properties of the set of numbers in the interval $[0,\frac{r}{s-1}]$ that admit a unique representation $x={\sum }_{n=1}^{\mathrm{\infty }}\frac{{\alpha }_n}{s^n}\equiv {\mathrm{\Delta }}_{{\alpha }_1{\alpha }_2\mathrm{.}\mathrm{.}\mathrm{.}{\alpha }_n\mathrm{.}\mathrm{.}\mathrm{.}}^{r_s}$, ${\alpha }_n\in A_r$. 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.