Fourier dimension of Mandelbrot multiplicative cascades

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 $${\dim }_F\mu =\min \{2,{\dim }_2\mu \},$$ 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 $${\dim }_F\mu \ge \frac{{\dim }_2\mu }{2+{\dim }_2\mu }\mathrm{.}$$