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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.03949 |
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| _version_ | 1866912807884161024 |
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| author | Pratsiovytyi, Mykola Vynnyshyn, Oleh |
| author_facet | Pratsiovytyi, Mykola Vynnyshyn, Oleh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet Pratsiovytyi, Mykola Vynnyshyn, Oleh Number Theory 11A67, 28A80, 28E15 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. |
| title | Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet |
| topic | Number Theory 11A67, 28A80, 28E15 |
| url | https://arxiv.org/abs/2601.03949 |