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arXiv · 2507.21605

On the Fourier transform of random Bernoulli convolutions

Simon Baker·Henna Koivusalo·Sascha Troscheit·Xintian Zhang
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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.

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Simon Baker, Henna Koivusalo, Sascha Troscheit, Xintian Zhang. 2025-07-29. On the Fourier transform of random Bernoulli convolutions. https://arxiv.org/abs/2507.21605

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