The exact dimension of Liouville numbers: The Fourier side

Abstract

In this article we study the generalized Fourier dimension of the set of Liouville numbers $L$. 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 $L$. 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 $L$ relying on its translation invariance property.