Fourier decay of Gaussian multiplicative chaos and boundary geometry

Abstract

In this paper, we develop a new approach to studying the Fourier transform of Gaussian multiplicative chaos. We determine the Fourier dimension of these random measures on tori for every smooth log-correlated covariance and in every dimension, and prove sharp bounds and exact formulas for chaos on natural classes of bounded sets and hypersurfaces in Euclidean space. These results highlight how boundary geometry and multifractal concentration jointly govern Fourier decay, including regimes in which the boundary strictly reduces the Fourier dimension. In the circle setting, this gives a substantially simpler new proof of a conjecture of Garban and Vargas.