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

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 $r_a$--representatio