Numeral systems with non-zero redundancy and their applications in the theory of locally complex functions
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arXiv
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2026
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| author | Vaskevych, S. O. Vovk, Yu. Yu. Pratsiovytyi, O. M. |
| author_facet | Vaskevych, S. O. Vovk, Yu. Yu. Pratsiovytyi, O. M. |
| contents | In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{α_n}{a^n}\equiv Δ^{r_a}_{α_1α_2...α_n...}, α_n\in A.\] Since the alphabet is redundant the numbers from the interval $[0;\frac{r}{a-1}]$ have not a single representation and can even have a continuous set of different representations.
We describe the geometry (topological and metric properties) of such representations (the $r_a$-representations) in terms of cylinders defined by \[Δ^{r_a}_{c_1c_2...c_m}= \{x: x=Δ^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\},\] We analyze their properties in detail, including the specific nature of overlaps.
We present results on the structural, variational, topological, metric and partially fractal properties of the function defined by \[f\left(x=\sum_{n=1}^{\infty}\frac{α_n}{(r+1)^n}\right)= Δ^{r_a}_{α_1α_2...α_n...},α_n \in A.\] We prove the function is continuous at all points of the interval $[0,1]$ that have a unique representation in the classical numeral system on the base $r+1$ and prove the function is discontinuous at points of a countable everywhere dense set in $[0,1]$. Furthermore, we show that the function is nowhere monotonic and has unlimited variation.
In the particular case $r=1$ and $a=\frac{1+\sqrt{5}}{2}$, we specify fractal level sets with Hausdorff--Besicovitch dimension not less than $-\log_a2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28606 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Numeral systems with non-zero redundancy and their applications in the theory of locally complex functions Vaskevych, S. O. Vovk, Yu. Yu. Pratsiovytyi, O. M. Number Theory Functional Analysis 26A21, 26A30 In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{α_n}{a^n}\equiv Δ^{r_a}_{α_1α_2...α_n...}, α_n\in A.\] Since the alphabet is redundant the numbers from the interval $[0;\frac{r}{a-1}]$ have not a single representation and can even have a continuous set of different representations. We describe the geometry (topological and metric properties) of such representations (the $r_a$-representations) in terms of cylinders defined by \[Δ^{r_a}_{c_1c_2...c_m}= \{x: x=Δ^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\},\] We analyze their properties in detail, including the specific nature of overlaps. We present results on the structural, variational, topological, metric and partially fractal properties of the function defined by \[f\left(x=\sum_{n=1}^{\infty}\frac{α_n}{(r+1)^n}\right)= Δ^{r_a}_{α_1α_2...α_n...},α_n \in A.\] We prove the function is continuous at all points of the interval $[0,1]$ that have a unique representation in the classical numeral system on the base $r+1$ and prove the function is discontinuous at points of a countable everywhere dense set in $[0,1]$. Furthermore, we show that the function is nowhere monotonic and has unlimited variation. In the particular case $r=1$ and $a=\frac{1+\sqrt{5}}{2}$, we specify fractal level sets with Hausdorff--Besicovitch dimension not less than $-\log_a2$. |
| title | Numeral systems with non-zero redundancy and their applications in the theory of locally complex functions |
| topic | Number Theory Functional Analysis 26A21, 26A30 |
| url | https://arxiv.org/abs/2603.28606 |