Multifractal behavior of polynomial Fourier series

TL;DR

This paper investigates the multifractal properties of Fourier series with polynomial frequencies by establishing bounds on the spectrum of singularities, which describes the distribution of local regularities across the function.

Contribution

It provides the first non-trivial bounds for the spectrum of singularities of such Fourier series, advancing understanding of their multifractal structure.

Findings

Established upper bounds for the spectrum of singularities.

Established lower bounds for the spectrum of singularities.

Characterized the Hausdorff dimension of sets with specific Hölder exponents.

Abstract

We prove non-trivial upper and lower bounds for the "Spectrum of Singularities" of Fourier Series with polynomial frequencies. The Spectrum of Singularities of a function f gives the Hausdorff dimension of the set of points with a given H\"older exponent for f.