{"paper":{"title":"On the Fourier transform of random Bernoulli convolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Henna Koivusalo, Sascha Troscheit, Simon Baker, Xintian Zhang","submitted_at":"2025-07-29T09:03:25Z","abstract_excerpt":"We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution\n  \\[\n  \\mu_\\omega = \\mathop{\\circledast}_{k=1}^{\\infty} \\left( \\frac{\\delta_0 + \\delta_{\\lambda_1 \\lambda_2 \\ldots \\lambda_{k-1} \\lambda_k}}{2} \\right),\n  \\] 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 sur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.21605","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.21605/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}