Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus

TL;DR

This paper precisely determines the Fourier dimensions of Gaussian Multiplicative Chaos measures on high-dimensional tori, resolving a longstanding open problem through advanced decomposition and multi-resolution analysis techniques.

Contribution

It introduces a novel construction of log-correlated Gaussian fields with specific decompositions, enabling exact Fourier dimension calculations in all dimensions.

Findings

Exact Fourier dimensions for all dimensions $d \\ge 1$

New decomposition method for Gaussian fields

Sharp local Fourier decay estimates

Abstract

We determine the exact values of the Fourier dimensions for Gaussian Multiplicative Chaos measures on the $d$-dimensional torus $\mathbb{T}^d$ for all integers $d \ge 1$. This resolves a problem left open in previous works [LQT24,LQT25] for high dimensions $d\ge 3$. The proof relies on a new construction of log-correlated Gaussian fields admitting specific decompositions into smooth processes with high regularity. This construction enables a multi-resolution analysis to obtain sharp local estimates on the measure's Fourier decay. These local estimates are then integrated into a global bound using Pisier's martingale type inequality for vector-valued martingales.