H\"older continuity and dimensions of fractal Fourier series

TL;DR

This paper investigates the regularity and fractal dimensions of Fourier series with applications to number theory and fractal geometry, establishing bounds on H"older continuity and graph dimensions under certain conditions.

Contribution

It provides new bounds on the H"older exponents and fractal dimensions of Fourier series, including classical functions and series involving the Möbius function, under general and RH-based assumptions.

Findings

Series are H"older continuous with exponent 1−α under certain bounds.

The graph of |F(t)| has box-counting dimension ≤ 1+α.

Applications include optimal regularity for Weierstrass and Riemann functions.

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.