The exact dimension of Liouville numbers: The Fourier side

TL;DR

This paper investigates the Fourier dimension of Liouville numbers, providing a detailed characterization of decay rates of Fourier transforms of measures supported on this zero Hausdorff dimension set, linking Fourier analysis with fractal geometry.

Contribution

It offers a near-complete classification of admissible Fourier decay rates for measures on Liouville numbers, connecting Fourier dimension with generalized Hausdorff dimension analysis.

Findings

Characterization of decay rates for Fourier transforms on Liouville numbers

Link between Fourier dimension and generalized Hausdorff dimension

Method to classify oscillating Fourier decay candidates

Abstract

In this article we study the generalized Fourier dimension of the set of Liouville numbers $\mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as control for the Fourier transform of (Rajchman) measures supported on $\mathbb{L}$. We give an almost complete characterization of admissible decays for this set in terms of comparison to power-like functions. This work can be seen as the ``Fourier side'' of the analysis made by Olsen and Renfro regarding the generalized Hausdorff dimension using gauge functions. We also provide an approach to deal with the problem of classifying oscillating candidates for a Fourier decay for $\mathbb{L}$ relying on its translation invariance property.