English

One continuum class of fractal functions defined in terms of $Q^*_s$-representation

Number Theory 2026-03-31 v1

Abstract

In the paper we study a class FF of multiparameter functions defined in terms of a polybasic ss-adic QsQ^{*}_{s}-representation of numbers by \begin{equation*} f_a\bigl(x=\Delta^{Q^{*}_s}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}\bigr) = \Delta^{Q^{*}s}_{|a_1-\alpha_1|\,|a_2-\alpha_2|\,\ldots\,|a_n-\alpha_n|\ldots}, \end{equation*} where (an)(a_n) is the sequence of digits for ss-adic representation of the parameter a[0,1]a\in[0,1], and \begin{equation*} \Delta^{Q^{*}_s}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}= \beta_{\alpha_1 1}+ \sum_{n=2}^{\infty} \left( \beta_{\alpha_n n} \prod_{j=1}^{n-1} q_{\alpha_j j} \right) \end{equation*} is the QsQ^{*}_{s}-representation of real numbers generated by a positive stochastic matrix qij\|q_{ij}\| with βαnn=i=0αn1qin\beta_{\alpha_n n}=\sum\limits_{i=0}^{\alpha_n-1} q_{in}. In this paper we investigate the continuity of the function faf_a on the sets of QsQ^{*}_{s}-binary and QsQ^{*}_{s}-unary numbers. We prove that the functions in this class are continuous on the set of numbers with a unique QsQ^{*}_{s}-representation. Furthermore, we show that except for f0f_0 and f1f_1, all functions have a countable set of discontinuities at QsQ^{*}_{s}-binary points. We classify the topological types of the value sets of faf_a depending on the parameter aa. We prove that, if the value set is of Cantor type, then it is zero-dimensional. We describe the structural properties of the level sets of faf_a in terms of the digits of the ss-adic representation of aa. In particular, we establish that a level set of the function faf_a can be an empty set, a finite set, or a continuum. For certain values of ss we provide examples of fractal level sets and calculate its fractal dimensions.

Keywords

Cite

@article{arxiv.2603.28598,
  title  = {One continuum class of fractal functions defined in terms of $Q^*_s$-representation},
  author = {V. V. Nazarchuk and S. O. Vaskevych and S. P. Ratushniak},
  journal= {arXiv preprint arXiv:2603.28598},
  year   = {2026}
}