On the Fourier transform of random Bernoulli convolutions

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution $${\mu }_{\omega }={}_{k=1}^{\mathrm{\infty }}\left(\frac{{\delta }_0+{\delta }_{{\lambda }_1{\lambda }_2\dots {\lambda }_{k-1}{\lambda }_k}}{2}\right),$$ where $\omega =({\lambda }_k)$ is a sequence of i.i.d. random variables each following the uniform distribution on some fixed interval. We study the regularity of these measures and prove that when $\exp \mathbb{E}(\log {\lambda }_1)>\frac{2}{\pi },$ the Fourier transform ${\widehat{\mu }}_{\omega }$ is an $L^1$ function almost surely. This in turn implies that the corresponding random self-similar set supporting ${\mu }_{\omega }$ has non-empty interior almost surely. This improves upon a previous bound due to Peres, Simon and Solomyak. Furthermore, under no assumptions on the value of $\exp \mathbb{E}(\log {\lambda }_1),$ we prove that ${\widehat{\mu }}_{\omega }$ will decay to zero at a polynomial rate almost surely.