H\"older continuity and dimensions of fractal Fourier series

Abstract

Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2\pi i nt}/n$, for a large class of coefficient functions $f$. Our main result states that if, for some constants $C$ and $\alpha$ with $0<\alpha<1$, we have $|\sum_{1\le n\le x}f(n)e^{2\pi i nt}|\le C x^{\alpha}$ uniformly in $x\ge 1$ and $t\in \mathbb{R}$, then the series $F(t)$ is H\"older continuous with exponent $1-\alpha$, and the graph of $|F(t)|$ on the interval $[0,1]$ has box-counting dimension $\leq 1+\alpha$. As applications we recover the best-possible uniform H\"older exponents for the Weierstrass functions $\sum_{k=1}^\infty a^k\cos(2\pi b^k t)$ and the Riemann function $\sum_{n=1}^\infty \sin(\pi n^2 t)/n^2$. Moreoever, under the assumption of the Generalized Riemann Hypothesis, we obtain nontrivial bounds for H\"older exponents and dimensions associated with series of the form $\sum_{n=1}^\infty \mu(n)e^{2\pi i n^kt}/n^k$, where $\mu$ is the M\"obius function.