Fourier dimension of Mandelbrot multiplicative cascades

TL;DR

This paper determines the Fourier dimension of Mandelbrot multiplicative cascade measures. For cascades generated by a sub-exponential random variable $W$, it proves $\dim_F\mu=\min\{2,\dim_2\mu\}$ with $\dim_2\mu=\alpha(W)$ given by a piecewise multifractal formula, and provides a parallel lower bound for cascades on the circle: $\dim_F\mu\ge\frac{\dim_2\mu}{2+\dim_2\mu}$. The lower bound is established via a SI-martingale analysis and Bernstein-type control of Fourier increments, while the upper bound employs projection arguments to rule out smooth densities of projections if the Fourier dimension exceeded 2. The circle case is treated with a van der Corput-type estimate to bound the Fourier transform and yield sharp interval bounds $\frac{\alpha}{2+\alpha}\le \dim_F\mu\le \alpha$. Overall, the work links Fourier decay to multifractal structure in Mandelbrot cascades and extends explicit Fourier-dimension results to spherical cascades, enriching the understanding of random fractal measures.

Abstract

We investigate the Fourier dimension, $\dim_Fμ$, of Mandelbrot multiplicative cascade measures $μ$ on the $d$-dimensional unit cube. We show that if $μ$ is the cascade measure generated by a sub-exponential random variable then \[\dim_Fμ=\min\{2,\dim_2μ\}\,,\] where $\dim_2μ$ is the correlation dimension of $μ$ and it has an explicit formula. For cascades on the circle $S\subset\mathbb{R}^2$, we obtain \[\dim_Fμ\ge\frac{\dim_2μ}{2+\dim_2μ}\,.\]

Theorems & Definitions (20)

• Remark 1.1
• Theorem 1.2
• Example 1.3
• proof
• Example 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• Lemma 2.2
• proof