  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.
