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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","authors_text":"Nicolas de Saxc\\'e, Pablo Shmerkin, Tuomas Orponen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2023-09-06T15:14:01Z","title":"On the Fourier decay of multiplicative convolutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.03068","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:14e8f782a9e21b1bcd60f0ca305d6668b36350a65eed9af6b666fc9b5f42a260","target":"record","created_at":"2026-07-05T07:49:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e4320cdd164a61453b71b4b8547350664ad6aada2c0f5777738ba9727a412462","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2023-09-06T15:14:01Z","title_canon_sha256":"43b2cb19a15084434df814f798413faaa8e67f9e5f0d391ff9284eec6b1f6d37"},"schema_version":"1.0","source":{"id":"2309.03068","kind":"arxiv","version":2}},"canonical_sha256":"f76e6526c31a48de75a305f468f33c6c2b38fa169ae45ea638ac92eff7606710","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f76e6526c31a48de75a305f468f33c6c2b38fa169ae45ea638ac92eff7606710","first_computed_at":"2026-07-05T07:49:55.532598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:49:55.532598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ERfZ/LqPeXykmzSgBw9cjvsyxrmhWUpsJgNJRw4ER7PaM4R/qe3kx+3pqW1vh+8Lk+zZI+KPjYvzaAHpcFfLCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T07:49:55.533084Z","signed_message":"canonical_sha256_bytes"},"source_id":"2309.03068","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14e8f782a9e21b1bcd60f0ca305d6668b36350a65eed9af6b666fc9b5f42a260","sha256:2658abfba5130f64d9f8d814846ecdf45c57b1972208d02ed5ea04536a3133d8"],"state_sha256":"a439181407236790d59ef35de7f19956b182367bceaa6ed3ea203e0c8d6e1e6b"}