Hölder continuity and dimensions of fractal Fourier series (2208.09806v4)

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.