The Fourier spectrum and sumset type problems

TL;DR

This paper introduces the Fourier spectrum, a new dimension concept bridging Fourier and Hausdorff dimensions, and applies it to sumset problems and distance set issues in harmonic analysis.

Contribution

It defines the Fourier spectrum, develops its fundamental properties, and demonstrates its utility in solving sumset and distance set problems under Fourier analytic conditions.

Findings

Fourier spectrum interpolates between Fourier and Hausdorff dimensions.

Established foundational theory for the Fourier spectrum.

Applied the concept to solve specific sumset and distance set problems.

Abstract

We introduce and study the \emph{Fourier spectrum} which is a continuously parametrised family of dimensions living between the Fourier dimension and the Hausdorff dimension for both sets and measures. We establish some fundamental theory and motivate the concept via several applications, especially to sumset type problems. For example, we study dimensions of convolutions and sumsets, and solve the distance set problem for sets satisfying certain Fourier analytic conditions.