On the Fourier dimension and a modification

TL;DR

This paper investigates the stability properties of Fourier dimension for sets and measures, providing conditions, counterexamples, and a modified definition to achieve countable stability.

Contribution

It introduces a sufficient condition for Fourier dimension stability, demonstrates non-stability in general, and proposes a modified Fourier dimension concept for improved stability.

Findings

Fourier dimension of a union equals the supremum of individual dimensions under certain conditions

Fourier dimension is not countably stable in general for sets and measures

A modified Fourier dimension can be defined to ensure countable stability

Abstract

We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that a measure has modified Fourier dimension greater than or equal to $s$ if and only if it annihilates all sets with modified Fourier dimension less than $s$.