The Fourier spectrum and sumset type problems
Classical Analysis and ODEs
2026-05-28 v4 Dynamical Systems
Functional Analysis
Metric Geometry
Probability
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.
Cite
@article{arxiv.2210.07019,
title = {The Fourier spectrum and sumset type problems},
author = {Jonathan M. Fraser},
journal= {arXiv preprint arXiv:2210.07019},
year = {2026}
}
Comments
31 pages, 1 figure, small updates