Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions

Source abstract

In this paper, we study a numeration system with a non--integer base $a>1$ over the binary alphabet $A=\{0,1\}$: $$\left[0;\frac{r}{a-1}\right]\ni x=\sum\limits_{n=1}^{\infty}\frac{α_n}{a^n} \equiv Δ^{r_a}_{α_1α_2\ldotsα_n\ldots}, \quad α_n\in A.$$ We investigate the geometry of $r_a$--representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by $$f\left(x=\sum\limits_{n=1}^{\infty}\frac{α_n}{2^n}\right) =\sum\limits_{n=1}^{\infty}\frac{α_n}{a^n},$$ where the classical binary representation of the argument is assumed not to end with the infinite period $(1)$. We derive a system of two functional equations satisfied by $f$. We prove that the function has fractal level sets and a self--affine graph. We also evaluate the integral of $f$ over the interval $[0,1]$.