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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2309.03068","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2023-09-06T15:14:01Z","cross_cats_sorted":[],"title_canon_sha256":"43b2cb19a15084434df814f798413faaa8e67f9e5f0d391ff9284eec6b1f6d37","abstract_canon_sha256":"e4320cdd164a61453b71b4b8547350664ad6aada2c0f5777738ba9727a412462"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:49:55.533084Z","signature_b64":"ERfZ/LqPeXykmzSgBw9cjvsyxrmhWUpsJgNJRw4ER7PaM4R/qe3kx+3pqW1vh+8Lk+zZI+KPjYvzaAHpcFfLCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f76e6526c31a48de75a305f468f33c6c2b38fa169ae45ea638ac92eff7606710","last_reissued_at":"2026-07-05T07:49:55.532598Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:49:55.532598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2309.03068","created_at":"2026-07-05T07:49:55.532658+00:00"},{"alias_kind":"arxiv_version","alias_value":"2309.03068v2","created_at":"2026-07-05T07:49:55.532658+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.03068","created_at":"2026-07-05T07:49:55.532658+00:00"},{"alias_kind":"pith_short_12","alias_value":"65XGKJWDDJEN","created_at":"2026-07-05T07:49:55.532658+00:00"},{"alias_kind":"pith_short_16","alias_value":"65XGKJWDDJEN45ND","created_at":"2026-07-05T07:49:55.532658+00:00"},{"alias_kind":"pith_short_8","alias_value":"65XGKJWD","created_at":"2026-07-05T07:49:55.532658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2508.19047","citing_title":"Furstenberg set theorem for transversal families of functions","ref_index":14,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ","json":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ.json","graph_json":"https://pith.science/api/pith-number/65XGKJWDDJEN45NDAX2GR4Z4NQ/graph.json","events_json":"https://pith.science/api/pith-number/65XGKJWDDJEN45NDAX2GR4Z4NQ/events.json","paper":"https://pith.science/paper/65XGKJWD"},"agent_actions":{"view_html":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ","download_json":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ.json","view_paper":"https://pith.science/paper/65XGKJWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2309.03068&json=true","fetch_graph":"https://pith.science/api/pith-number/65XGKJWDDJEN45NDAX2GR4Z4NQ/graph.json","fetch_events":"https://pith.science/api/pith-number/65XGKJWDDJEN45NDAX2GR4Z4NQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ/action/storage_attestation","attest_author":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ/action/author_attestation","sign_citation":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ/action/citation_signature","submit_replication":"https://pith.science/pith/65XGKJWDDJEN45NDAX2GR4Z4NQ/action/replication_record"}},"created_at":"2026-07-05T07:49:55.532658+00:00","updated_at":"2026-07-05T07:49:55.532658+00:00"}