On the Fourier transform of random Bernoulli convolutions

TL;DR

This paper studies the Fourier transform properties of random Bernoulli convolutions, showing under certain conditions it is integrable and the support has non-empty interior, with results on decay rates without restrictions.

Contribution

It proves that for certain parameters, the Fourier transform of these convolutions is in L^1 and the support has interior, improving previous bounds and establishing decay rates without restrictions.

Findings

Fourier transform is in L^1 almost surely under specific conditions.

The support of the measure has non-empty interior almost surely.

The Fourier transform decays polynomially at infinity almost surely.

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.