On the Fourier transform of random Bernoulli convolutions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915428017635328 |
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| author | Baker, Simon Koivusalo, Henna Troscheit, Sascha Zhang, Xintian |
| author_facet | Baker, Simon Koivusalo, Henna Troscheit, Sascha Zhang, Xintian |
| contents | 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 $ω=(λ_k)$ 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 λ_1\right)>\frac{2}π, $ the Fourier transform $\widehatμ_ω$ is an $L^{1}$ function almost surely. This in turn implies that the corresponding random self-similar set supporting $μ_ω$ 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 λ_1), $ we prove that $\widehat μ_ω$ will decay to zero at a polynomial rate almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Fourier transform of random Bernoulli convolutions Baker, Simon Koivusalo, Henna Troscheit, Sascha Zhang, Xintian Dynamical Systems Probability 37H12 (Primary) 42A38, 28A80 (Secondary) 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 $ω=(λ_k)$ 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 λ_1\right)>\frac{2}π, $ the Fourier transform $\widehatμ_ω$ is an $L^{1}$ function almost surely. This in turn implies that the corresponding random self-similar set supporting $μ_ω$ 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 λ_1), $ we prove that $\widehat μ_ω$ will decay to zero at a polynomial rate almost surely. |
| title | On the Fourier transform of random Bernoulli convolutions |
| topic | Dynamical Systems Probability 37H12 (Primary) 42A38, 28A80 (Secondary) |
| url | https://arxiv.org/abs/2507.21605 |