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 \[\dim_Fμ=\min\{2,\dim_2μ\}\,,\] 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 \[\dim_Fμ\ge\frac{\dim_2μ}{2+\dim_2μ}\,.\]

Revision based on a referee report. Changes to the structure of the paper and to the derivation of the correlation dimension. All main results remain the same