On one class of fractal sets

Abstract

In the present article a new class $\Upsilon$ of all sets represented in the following form is introduced: $$ \mathbb S_{(s,u)}\equiv\left\{x: x= \Delta^{s}_{{\underbrace{u...u}_{\alpha_1-1}} \alpha_1{\underbrace{u...u}_{\alpha_2 -1}}\alpha_2 ...{\underbrace{u...u}_{ \alpha_n -1}}\alpha_n...}, \alpha_n \in A_0, \alpha_n \ne u, \alpha_n \ne 0 \right\}, $$ where $2<s\in \mathbb N$ and $u\in A$ are fixed parameters, $A=\{0,1,\dots,s-1\}$, and $A_0=A \setminus \{0\}$. Topological, metric, and fractal properties of these sets are studied. The problem of belonging to these sets of normal numbers is investigated. The theorem on the calculation of the Hausdorff-Besicovitch dimension of an arbitrary set whose elements have restrictions on using of digits or combinations of digits in own s-adic representations is formulated and proved. In 2012, the results of this article were presented by the author in the abstracts (https://www.researchgate.net/publication/303054326, https://www.researchgate.net/publication/303053939, https://www.researchgate.net/publication/303054835). The investigation was represented at the seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine (29, September 2011 and 16, February 2012, available at http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1) and such investigations for the case of the nega-s-adic representation were published by the author in Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics and Mathematics 15 (2013), p. 168-187 (https://www.researchgate.net/publication/292970280, in Ukrainian).