Fourier dimension of random images

TL;DR

This paper constructs smooth diffeomorphisms that increase the Fourier dimension of measures and sets, showing that Fourier dimension can be manipulated via smooth transformations and is not invariant under certain diffeomorphisms.

Contribution

It introduces a method to increase Fourier dimension of measures and sets through smooth diffeomorphisms, demonstrating non-invariance of Fourier dimension under $C^m$-diffeomorphisms.

Findings

Every Borel set is diffeomorphic to a Salem set.

Fourier dimension can be made arbitrarily large relative to Hausdorff dimension.

Fourier dimension is not invariant under $C^m$-diffeomorphisms.

Abstract

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$. This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$-equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha)$. In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^m$-diffeomorphisms for any $m$.