Fourier Dimension of Mandelbrot Multiplicative Cascades

Abstract

We investigate the Fourier dimension, \(\dim _F\mu \), of Mandelbrot multiplicative cascade measures \(\mu \) on the d-dimensional unit cube. We show that if \(\mu \) is the cascade measure generated by a sub-exponential random variable, then

$$\begin{aligned} \dim _F\mu =\min \{2,\dim _2\mu \}, \end{aligned}$$

where \(\dim _2\mu \) is the correlation dimension of \(\mu \) and it has an explicit formula. For cascades on the circle \(S\subset \mathbb {R}^2\), we obtain

$$\begin{aligned} \dim _F\mu \ge \frac{\dim _2\mu }{2+\dim _2\mu }. \end{aligned}$$