Hölder 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 $F(t)={\sum }_{n=1}^{\infty }f(n)e^{2\pi int}/n$, 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 $|{\sum }_{1\le n\le x}f(n)e^{2\pi int}|\le C{x}^{\alpha }$ uniformly in $x\ge 1$ and $t\in \mathbb{R}$, then the series $F(t)$ is Hölder continuous with exponent $1-\alpha$, and the graph of $|F(t)|$ on the interval $[0,1]$ has box-counting dimension $\le 1+\alpha$. As applications we recover the best-possible uniform Hölder 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 Hölder exponents and dimensions associated with series of the form ${\sum }_{n=1}^{\infty }\mu (n)e^{2\pi i{n}^{k}t}/n^k$, where $\mu$ is the Möbius function.

19 pages, 5 figures, References added, To appear in J Anal