A Dichotomy for the Weierstrass-type functions

TL;DR

This paper establishes a clear dichotomy for Weierstrass-type functions generated by real analytic periodic functions, showing they are either analytic or have graphs with a specific Hausdorff dimension, with only finitely many exceptions.

Contribution

It proves a dichotomy for Weierstrass-type functions based on the parameters, linking analyticity to Hausdorff dimension and characterizing when each case occurs.

Findings

Either the function is real analytic or its graph has Hausdorff dimension 2 + log_b(λ).

The analytic case occurs only for finitely many λ unless φ is constant.

The result characterizes the structure of Weierstrass-type functions based on parameters.

Abstract

For a real analytic periodic function $\phi:\mathbb{R}\to \mathbb{R}$, an integer $b\ge 2$ and $\lambda\in (1/b,1)$, we prove the following dichotomy for the Weierstrass-type function $W(x)=\sum\limits_{n\ge 0}{{\lambda}^n\phi(b^nx)}$: Either $W(x)$ is real analytic, or the Hausdorff dimension of its graph is equal to $2+\log_b\lambda$. Furthermore, given $b$ and $\phi$, the former alternative only happens for finitely many $\lambda$ unless $\phi$ is constant.