Fractal functions defined in terms of number representations in systems with a redundant alphabet
M. V. Pratsiovytyi, S. P. Ratushniak, Yu. Yu. Vovk, Ya. V. Goncharenko
TL;DR
The paper analyzes numeral systems with base $s$ and redundant alphabet $A_r$, introducing the $r_s$-expansion and proving constructive existence for all $x\in[0,\frac{r}{s-1}]$. It builds a geometric framework based on cylinder sets and overlaps and defines a central map $f$ that links the $(r+1)$-ary and $r_s$-representations, establishing continuity properties, unbounded variation, and a self-affine graph. The work derives a functional system and a precise self-affine dimension for the graph, and computes the average value of $f$ over $[0,1]$. Finally, it classifies the level sets of $f$ by the relation between $r$ and $s$, showing fractal or anomalously fractal behavior and giving explicit Hausdorff dimensions for key regimes.
Abstract
For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion $x=\sum\limits_{n=1}^{\infty}s^{-n}α_n=Δ^{r_s}_{α_1α_2...α_n...}.$ The algorithm for expanding a number into such a series is justified in the paper. The geometry of this representation is studied, including the geometric meaning of digits, properties of cylinder sets -- particularly the specificity of their overlaps -- and metric relations, as well as the connection between the representation and partial sums of the corresponding series. The paper also presents results on the study of a function $f$ defined by $f(x=\sum\limits_{n=1}^{\infty}\frac{α_n}{(r+1)^n})=Δ^{r_s}_{α_1α_2...α_n...}, α_n\in A.$ It is proved that the function $f$ is continuous at every point that has a unique representation in the classical numeration system with base $r+1$, and discontinuous at points having two representations. The function has unbounded variation and a self-affine graph. For $r<2s-1$, the function possesses singleton, finite, countable, and continuum level sets, including fractal ones; for $r>2s-2$, every level set is a continuum, and moreover it is fractal or anomalously fractal.