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

Abstract: In this paper, we study a numeration system with a non--integer base $a&gt;1$ over the binary alphabet $A={0,1}$: $$\left[0;\frac{r}{a-1}\right]\ni x=\sum\limits_{n=1}^{{\infty}\frac{α_n}{a^n}} \equiv Δ^{{r_a}_{α_1α_2\ldotsα_n\ldots},} \quad α<em>n\in A.$$ 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 $$f\left(x=\sum\limits</em>{n=1}^{{\infty}\frac{α_n}{2^n}\right)} =\sum\limits_{n=1}<sup>{\infty}\frac{α_n}{a<sup>n},$$ 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 $[0,1]$.