Fractal functions defined in terms of number representations in systems with a redundant alphabet

TL;DR

This paper explores fractal functions derived from number representations in systems with redundant alphabets, analyzing their geometric properties, continuity, and level set structures, revealing complex fractal behaviors.

Contribution

It introduces a new representation of numbers using redundant alphabets, studies the associated fractal functions, and characterizes their continuity and level set properties.

Findings

The function is continuous at points with unique representations.

The function is discontinuous at points with multiple representations.

Level sets exhibit fractal and anomalous fractal structures.

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 $x=\sum\limits_{n=1}^{\infty}s^{-n}\alpha_n=\Delta^{r_s}_{\alpha_1\alpha_2...\alpha_n...}.$ 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 $f(x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{(r+1)^n})=\Delta^{r_s}_{\alpha_1\alpha_2...\alpha_n...}, \alpha_n\in A.$ 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.