Numeral systems with non-zero redundancy and their applications in the theory of locally complex functions

TL;DR

This paper investigates numeral systems with non-zero redundancy, analyzing their geometric, topological, and fractal properties, and studies the continuity, discontinuity, and variation of associated functions.

Contribution

It provides a detailed analysis of the properties of redundant numeral representations and the functions they define, including their fractal and topological characteristics.

Findings

The function is continuous at points with unique classical representations.

The function is discontinuous at a dense set of points in [0,1].

The function exhibits unlimited variation and is nowhere monotonic.

Abstract

In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n}\equiv \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...}, \alpha_n\in A.\] Since the alphabet is redundant the numbers from the interval $[0;\frac{r}{a-1}]$ have not a single representation and can even have a continuous set of different representations. We describe the geometry (topological and metric properties) of such representations (the $r_a$-representations) in terms of cylinders defined by \[\Delta^{r_a}_{c_1c_2...c_m}= \{x: x=\Delta^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\},\] We analyze their properties in detail, including the specific nature of overlaps. We present results on the structural, variational, topological, metric and partially fractal properties of the function defined by \[f\left(x=\sum_{n=1}^{\infty}\frac{\alpha_n}{(r+1)^n}\right)= \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...},\alpha_n \in A.\] We prove the function is continuous at all points of the interval $[0,1]$ that have a unique representation in the classical numeral system on the base $r+1$ and prove the function is discontinuous at points of a countable everywhere dense set in $[0,1]$. Furthermore, we show that the function is nowhere monotonic and has unlimited variation. In the particular case $r=1$ and $a=\frac{1+\sqrt{5}}{2}$, we specify fractal level sets with Hausdorff--Besicovitch dimension not less than $-\log_a2$.