Hausdorff dimension of level sets of generalized Takagi functions

TL;DR

This paper investigates the Hausdorff dimension of level sets of generalized Takagi functions, providing sharp bounds and analyzing the dimensions of zero and maximum point sets for randomly chosen functions.

Contribution

It introduces sharp bounds for the Hausdorff dimension of level sets of generalized Takagi functions and studies the dimensions of specific point sets for random functions.

Findings

Sharp upper bounds for Hausdorff dimensions of level sets

Dimensions of zero and maximum point sets for random functions

Extension of Takagi function analysis to broader families

Abstract

This paper examines level sets of two families of continuous, nowhere differentiable functions (one a subfamily of the other) defined in terms of the "tent map". The well-known Takagi function is a special case. Sharp upper bounds are given for the Hausdorff dimension of the level sets of functions in these two families. Furthermore, the case where a function f is chosen at random from either family is considered, and results are given for the Hausdorff dimension of the zero set and the set of maximum points of f.