Topological, metric and fractal properties of the set of real numbers with a given asymptotic mean of digits in their $4$-adic representation in the case when the digit frequencies exist
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arXiv
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| Natura: | Preprint |
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2026
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| author | Pratsiovytyi, M. V. Klymchuk, S. O. |
| author_facet | Pratsiovytyi, M. V. Klymchuk, S. O. |
| contents | In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}α_k(x), \text{ where } x=\sum^{\infty}_{k=1}α_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of real number $x$, particularly properties of it's level sets $ S_θ=\left\{x: r(x)=θ,\: θ=const, \: 0\leqslantθ\leqslant 3\right\}, $ if all $4$-adic digits frequencies exist, i.e. $$ ν_i(x)=\lim_{n\to\infty}n^{-1}\#\{k: α_k(x)=i, i\leqslant n\}, \:\: i=0,1,2,3. $$ We provided an algorithm of constructing point from the set $S_θ$, and proved continuality and every where density of the set. We found conditions of zero and full Lebesgue measure and estimates of Hausdorff-Besicovitch fractal dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_03399 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological, metric and fractal properties of the set of real numbers with a given asymptotic mean of digits in their $4$-adic representation in the case when the digit frequencies exist Pratsiovytyi, M. V. Klymchuk, S. O. Number Theory Functional Analysis 11K55, 26A27, 26A30 In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}α_k(x), \text{ where } x=\sum^{\infty}_{k=1}α_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of real number $x$, particularly properties of it's level sets $ S_θ=\left\{x: r(x)=θ,\: θ=const, \: 0\leqslantθ\leqslant 3\right\}, $ if all $4$-adic digits frequencies exist, i.e. $$ ν_i(x)=\lim_{n\to\infty}n^{-1}\#\{k: α_k(x)=i, i\leqslant n\}, \:\: i=0,1,2,3. $$ We provided an algorithm of constructing point from the set $S_θ$, and proved continuality and every where density of the set. We found conditions of zero and full Lebesgue measure and estimates of Hausdorff-Besicovitch fractal dimension. |
| title | Topological, metric and fractal properties of the set of real numbers with a given asymptotic mean of digits in their $4$-adic representation in the case when the digit frequencies exist |
| topic | Number Theory Functional Analysis 11K55, 26A27, 26A30 |
| url | https://arxiv.org/abs/2603.03399 |