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

Анотація

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of num\-bers 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

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

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.