H\"older continuity and dimensions of fractal Fourier series

arXiv2208.09806MaRDI QIDQ6408379

Efstathios Konstantinos Chrontsios Garitsis, AJ Hildebrand

Publication date: 21 August 2022

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 $F(t)=su{m}_{n=1}^{i}nftyf(n)e^{2piint}/n$, 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 $|su{m}_{1lenlex}f(n)e^{2piint}|leC{x}^{alpha}$ uniformly in $xge1$ and $tinmathbbR$, 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 $leq1+alpha$. As applications we recover the best-possible uniform H"older exponents for the Weierstrass functions $su{m}_{k=1}^{i}nfty{a}^{k}cos(2pi{b}^{k}t)$ and the Riemann function $su{m}_{n=1}^{i}nftysin(pi{n}^{2}t)/n^2$. 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 $su{m}_{n=1}^{i}nftymu(n)e^{2pii{n}^{k}t}/n^k$, where $mu$ is the Moebius function.

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