On the Fourier transform of random Bernoulli convolutions

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution image where 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 , the Fourier transform is an function almost surely. This in turn implies that the corresponding random self‐similar set supporting 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 , we prove that will decay to zero at a polynomial rate almost surely.