On the Fourier transform of random Bernoulli convolutions

Abstract: We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution [ \mu_\omega = \mathop{\circledast}{k=1}^{\infty} \left( \frac{\delta_0 + \delta{\lambda_1 \lambda_2 \ldots \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 $\exp\mathbb{E}\left( \log \lambda_1\right)>\frac{2}{\pi},$ 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.