On the Fourier transform of random Bernoulli convolutions

Abstract

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ μ_ω= \mathop{\circledast}_{k=1}^{\infty} \left( \frac{δ_0 + δ_{λ_1 λ_2 \ldots λ_{k-1} λ_k}}{2} \right), \] where $\ddot{I}‰=(\hat{I}{»}_k)$ 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 $\exp \mathbb{E}\left(\log \hat{I}{»}_1\right)>\frac{2}{\ddot{I}}€,$ the Fourier transform $\hat{\hat{I}}{¼}_{\ddot{I}}‰$ is an $L^1$ function almost surely. This in turn implies that the corresponding random self-similar set supporting $\hat{I}{¼}_{\ddot{I}}‰$ 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 $\exp \mathbb{E}(\log \hat{I}{»}_1),$ we prove that $\hat{\hat{I}}{¼}_{\ddot{I}}‰$ will decay to zero at a polynomial rate almost surely.