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 $$\dim_F\mu\ge\frac{\dim_2\mu}{2+\dim_2\mu}\,.$$