A Dichotomy for the Weierstrass-type functions

Abstract

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

37 pages, typos correced and small improvements

References in corpus (1)

• An elementary proof for the dimension of the graph of the classical Weierstrass function