H\"older exponents and fractal structure of level sets of self-affine functions associated with the $Q_s$-representation of numbers

TL;DR

This paper studies the fractal and local regularity properties of a class of self-affine functions linked to the $Q_s$-representation of real numbers, revealing their complex structure and level set characteristics.

Contribution

It provides explicit calculations of local Hölder exponents and characterizes the fractal nature of level sets and maximum points of these functions.

Findings

Computed local Hölder exponents at points with specific digit frequencies.

Identified conditions for the existence of continuum level sets.

Described the fractal structure of the set of maximum points.

Abstract

We investigate a class of locally complicated self-affine functions defined via the $Q_s$-representation of real numbers. In particular, we compute local H\"older exponents at points with given asymptotic frequencies of digits in their $Q_s$-representation. Furthermore, we establish conditions under which these functions possess continuum level sets. Finally, for self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points and show that this set can be fractal.