Harmonic analysis of Mandelbrot cascades -- in the context of vector-valued martingales

TL;DR

This paper determines the Fourier dimension of Mandelbrot cascade measures, revealing phase transitions and regularity properties, by applying a novel vector-valued martingale approach to multiplicative chaos analysis.

Contribution

It introduces a new vector-valued martingale framework to analyze Fourier decay in multiplicative chaos measures, solving a longstanding open problem.

Findings

Exact Fourier dimension formula for MCCM with finite moments.

MCCM is Salem if and only if the weight distribution is two-point.

MCCM exhibits polynomial Fourier decay under certain moment conditions.

Abstract

We solve a long-standing open problem of determining the Fourier dimension of the Mandelbrot canonical cascade measure (MCCM). This problem of significant interest was raised by Mandelbrot in 1976 and reiterated by Kahane in 1993. Specifically, we derive the exact formula for the Fourier dimension of the MCCM for random weights $W$ satisfying the condition $\mathbb{E}[W^t]<\infty$ for all $t>0$. As a corollary, we prove that the MCCM is Salem if and only if the random weight has a specific two-point distribution. In addition, we show that the MCCM is Rajchman with polynomial Fourier decay whenever the random weight satisfies $\mathbb{E}[W^{1+\delta}]<\infty$ for some $\delta>0$. As a consequence, we discover that, in the Biggins-Kyprianou's boundary case, the Fourier dimension of the MCCM exhibits a second order phase transition at the inverse temperature $\beta = 1/2$; we establish the upper Frostman regularity for MCCM; and we obtain a Fourier restriction estimate for MCCM. The major novelty of this paper is the discovery of putting the fine analysis of Fourier decay for multiplicative chaos measures into the theory of vector-valued martingales. This new viewpoint is of fundamental importance in the study of Fourier decay of multiplicative chaos measures. Indeed, in the sequel to this paper, combining the vector-valued martingale methods and ideas from Littlewood-Paley theory, the precise Fourier dimensions will be established for various classical models of multiplicative chaos measures including GMC of all dimensions, microcanonical Mandelbrot cascades, Mandelbrot random coverings, as well as Fourier-Walsh analysis of these models.