H\"older regularity of arithmetic Fourier series arising from modular forms

Abstract

Given a modular form which is not a cusp form $M_k(z)=\sum_{n=0}^{\infty}r_ne^{2\pi inz}$ of weight $k \geq 4$, we define the series $M_{k,s}(x)=\sum_{n=1}^{\infty}\frac{r_n}{n^s}\sin(2\pi nx),$ which converges for all $x\in\mathbb{R}$ when $s>k$. In this paper, we compute the H\"{o}lder regularity exponent of $M_{k,s}$ at irrational points. In our analysis we apply wavelets methods proposed by Jaffard in 1996 in the study of the Riemann series. We find that the H\"{o}lder regularity exponent at a point $x$ is related to the fine diophantine properties of $x$, in a very precise way.