Harmonic analysis of multiplicative chaos Part II: a unified approach to Fourier dimensions

TL;DR

This paper develops a unified method to analyze the Fourier decay of multiplicative chaos measures, determining their Fourier dimensions for key models and establishing bounds for others, advancing understanding of their harmonic properties.

Contribution

It introduces a unified approach to study Fourier decay in multiplicative chaos, resolving conjectures and providing precise Fourier dimensions for several models.

Findings

Exact Fourier dimensions for 1D and 2D GMC measures

Fourier dimensions for Mandelbrot measures and cascades

Lower bounds for Fourier dimensions in various models

Abstract

We introduce a unified approach for studying the polynomial Fourier decay of classical multiplicative chaos measures. As consequences, we obtain the precise Fourier dimensions for multiplicative chaos measures arising from the following key models: the sub-critical 1D and 2D GMC (which in particular resolves the Garban-Vargas conjecture); the sub-critical $d$-dimensional GMC with $d \ge 3$ when the parameter $\gamma$ is near the critical value; the canonical Mandelbrot random coverings; the canonical Mandelbrot cascades. For various other models, we establish the non-trivial lower bounds of the Fourier dimensions and in various cases we conjecture that they are all optimal and provide the exact values of Fourier dimensions.