{"paper":{"title":"On the Fourier decay of multiplicative convolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Nicolas de Saxc\\'e, Pablo Shmerkin, Tuomas Orponen","submitted_at":"2023-09-06T15:14:01Z","abstract_excerpt":"We prove the following. Let $\\mu_{1},\\ldots,\\mu_{n}$ be Borel probability measures on $[-1,1]$ such that $\\mu_{j}$ has finite $s_j$-energy for certain indices $s_{j} \\in (0,1]$ with $s_{1} + \\ldots + s_{n} > 1$. Then, the multiplicative convolution of the measures $\\mu_{1},\\ldots,\\mu_{n}$ has power Fourier decay: there exists a constant $\\tau = \\tau(s_{1},\\ldots,s_{n}) > 0$ such that\n  \\[\n  \\left| \\int e^{-2\\pi i \\xi \\cdot x_{1}\\cdots x_{n}} \\, d\\mu_{1}(x_{1}) \\cdots \\, d\\mu_{n}(x_{n}) \\right| \\leq |\\xi|^{-\\tau}\n  \\]\n  for sufficiently large $|\\xi|$. This verifies a suggestion of Bourgain from"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.03068","kind":"arxiv","version":2},"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/2309.03068/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"}