A generalization of the Takagi function for beta-expansions

TL;DR

This paper introduces a generalized Takagi function for beta-expansions with base between 1 and 2, analyzing its regularity and oscillatory behavior related to digit sums and multifractal properties.

Contribution

It extends the classical Takagi function to beta-expansions, providing new insights into its Hölder continuity and oscillation characteristics for bases in (1,2].

Findings

The generalized Takagi function is pointwise α-Hölder continuous for all α in (0,1).

It is not pointwise Lipschitz continuous on the unit interval except on a null set.

The analysis relies on a formula capturing oscillations of digit sums and limit theorems for the beta-map.

Abstract

We consider a generalized Takagi function for beta-expansions with the base $1<\beta\leq2$, motivated by multifractal analysis for digit frequency sets of beta-expansions [20]. We show that it is pointwise $\alpha$-H\"older continuous for any $\alpha\in(0,1)$ but not pointwise Lipschitz continuous on the unit interval except a Lebesgue null set. Our proof relies on a formula for the generalized Takagi function reflecting its oscillations of the sum of digits and some basic limit theorems for the corresponding beta-map.