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Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \\geq 1$ such that $$\\|\\hat{\\sigma}\\|_{L^{p}(B(R))} \\leq C_{s,t}R^{(2 - t)/p}, \\qquad R \\geq 1.$$ Moreover, when $s \\geq 2/3$ and $t \\in [0,s + 1)$, the previous inequality is true for $p \\geq 6$.\n  We also obtain the following fractal geometric counterpart of the previous re"},"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":"2401.17867","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2024-01-31T14:26:13Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"d6f00b18c62586a91b11a87461d7d0c05429a0034bcfb9ec5e2014b4ea03e3b5","abstract_canon_sha256":"b0e156cd1bbd041ab146ea08f5fa499f4a406ae41b34b01e7280d9224ce4246a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:33:13.430682Z","signature_b64":"ic++Z7QX1rYU1EbS2mAfdBHmHN+1qKhtdRu/OdM5QD9v0mtVvMKgpM115cuYZ3SYa7//WWjaZ6tmOax+fE6IDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"787927646b73d8e8bde085592f0a7eb6793388e5370d909d2951c58115d8e4ab","last_reissued_at":"2026-07-05T11:33:13.430195Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:33:13.430195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Fourier transforms of fractal measures on the parabola","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Aleksi Py\\\"or\\\"al\\\"a, Carmelo Puliatti, Tuomas Orponen","submitted_at":"2024-01-31T14:26:13Z","abstract_excerpt":"Let $s \\in [0,1]$ and $t \\in [0,\\min\\{3s,s + 1\\})$. 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Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \\geq 1$ such that $$\\|\\hat{\\sigma}\\|_{L^{p}(B(R))} \\leq C_{s,t}R^{(2 - t)/p}, \\qquad R \\geq 1.$$ Moreover, when $s \\geq 2/3$ and $t \\in [0,s + 1)$, the previous inequality is true for $p \\geq 6$.\n  We also obtain the following fractal geometric counterpart of the previous re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.17867","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/2401.17867/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":"2401.17867","created_at":"2026-07-05T11:33:13.430248+00:00"},{"alias_kind":"arxiv_version","alias_value":"2401.17867v3","created_at":"2026-07-05T11:33:13.430248+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.17867","created_at":"2026-07-05T11:33:13.430248+00:00"},{"alias_kind":"pith_short_12","alias_value":"PB4SOZDLOPMO","created_at":"2026-07-05T11:33:13.430248+00:00"},{"alias_kind":"pith_short_16","alias_value":"PB4SOZDLOPMORPPA","created_at":"2026-07-05T11:33:13.430248+00:00"},{"alias_kind":"pith_short_8","alias_value":"PB4SOZDL","created_at":"2026-07-05T11:33:13.430248+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.07321","citing_title":"$L^2$-Flattening of Self-similar Measures on Non-degenerate Curves","ref_index":29,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ","json":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ.json","graph_json":"https://pith.science/api/pith-number/PB4SOZDLOPMORPPAQVMS6CT6WZ/graph.json","events_json":"https://pith.science/api/pith-number/PB4SOZDLOPMORPPAQVMS6CT6WZ/events.json","paper":"https://pith.science/paper/PB4SOZDL"},"agent_actions":{"view_html":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ","download_json":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ.json","view_paper":"https://pith.science/paper/PB4SOZDL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2401.17867&json=true","fetch_graph":"https://pith.science/api/pith-number/PB4SOZDLOPMORPPAQVMS6CT6WZ/graph.json","fetch_events":"https://pith.science/api/pith-number/PB4SOZDLOPMORPPAQVMS6CT6WZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ/action/storage_attestation","attest_author":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ/action/author_attestation","sign_citation":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ/action/citation_signature","submit_replication":"https://pith.science/pith/PB4SOZDLOPMORPPAQVMS6CT6WZ/action/replication_record"}},"created_at":"2026-07-05T11:33:13.430248+00:00","updated_at":"2026-07-05T11:33:13.430248+00:00"}