Harmonic analysis of multiplicative chaos Part I: the proof of Garban-Vargas conjecture for 1D GMC

TL;DR

This paper proves the Garban-Vargas conjecture for 1D Gaussian multiplicative chaos by determining the exact Fourier dimensions, using an improved vector-valued martingale method.

Contribution

It provides a rigorous proof of the Fourier dimensions of sub-critical Gaussian multiplicative chaos, confirming a longstanding conjecture.

Findings

Exact Fourier dimensions of 1D GMC established

Confirmation of Garban-Vargas conjecture for sub-critical case

Enhanced martingale method for Fourier analysis

Abstract

In this paper, we establish the exact Fourier dimensions of all standard sub-critical Gaussian multiplicative chaos on the unit interval, thereby confirming the Garban-Vargas conjecture. The proof relies on a significant improvement of the vector-valued martingale method, initially developed by Chen-Han-Qiu-Wang in the studies of the Fourier dimensions of Mandelbrot cascade random measures.