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We proved that if $f \\in L^p(\\mathbb{R}^d)$ with \\begin{align*}\n  1 \\leq p\\leq p_\\alpha:=\n  \\begin{cases}\n  \\frac{d^2+d+2\\alpha}{2\\alpha} & d \\geq 3,\n  \\frac{4}{\\alpha} &d =2,\n  \\end{cases}\n  \\end{align*} and $\\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2412.19956","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-27T23:52:25Z","cross_cats_sorted":[],"title_canon_sha256":"b27b1b175317ac3a645f7ad4218921ddfacd0133c215d8568d5681319db5bd3f","abstract_canon_sha256":"42643af02d4eb66e3b93c392a07943d7b515c1a453176ca82ddd776167b79adc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:05:11.831929Z","signature_b64":"djt67CJsXfoVwGQPh+8QG3He5HOoRWZybSC1cesv30E+xVuj+QB2l4tvV0eqq9j3qZuoLBJc0lz1+QbNgEPXCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6657c85df5359e693f52834d4a9cc0ca5358bfe94289841a5014345a270b81f6","last_reissued_at":"2026-07-05T12:05:11.831492Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:05:11.831492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Donggeun Ryou, Minh-Quy Pham, Shengze Duan","submitted_at":"2024-12-27T23:52:25Z","abstract_excerpt":"For $0 < \\alpha \\leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\\mathbb{R}^d$ such that $N(E,\\varepsilon) \\lesssim \\varepsilon^{-\\alpha}$ for $0 <\\varepsilon <1$ where $N(E,\\varepsilon)$ is the smallest number of $\\varepsilon$-balls needed to cover $E$. We proved that if $f \\in L^p(\\mathbb{R}^d)$ with \\begin{align*}\n  1 \\leq p\\leq p_\\alpha:=\n  \\begin{cases}\n  \\frac{d^2+d+2\\alpha}{2\\alpha} & d \\geq 3,\n  \\frac{4}{\\alpha} &d =2,\n  \\end{cases}\n  \\end{align*} and $\\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.19956","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/2412.19956/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":"2412.19956","created_at":"2026-07-05T12:05:11.831560+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.19956v2","created_at":"2026-07-05T12:05:11.831560+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.19956","created_at":"2026-07-05T12:05:11.831560+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZL4QXPVGWPG","created_at":"2026-07-05T12:05:11.831560+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZL4QXPVGWPGSP2S","created_at":"2026-07-05T12:05:11.831560+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZL4QXPV","created_at":"2026-07-05T12:05:11.831560+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ","json":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ.json","graph_json":"https://pith.science/api/pith-number/MZL4QXPVGWPGSP2SQNGUVHGAZJ/graph.json","events_json":"https://pith.science/api/pith-number/MZL4QXPVGWPGSP2SQNGUVHGAZJ/events.json","paper":"https://pith.science/paper/MZL4QXPV"},"agent_actions":{"view_html":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ","download_json":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ.json","view_paper":"https://pith.science/paper/MZL4QXPV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.19956&json=true","fetch_graph":"https://pith.science/api/pith-number/MZL4QXPVGWPGSP2SQNGUVHGAZJ/graph.json","fetch_events":"https://pith.science/api/pith-number/MZL4QXPVGWPGSP2SQNGUVHGAZJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ/action/storage_attestation","attest_author":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ/action/author_attestation","sign_citation":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ/action/citation_signature","submit_replication":"https://pith.science/pith/MZL4QXPVGWPGSP2SQNGUVHGAZJ/action/replication_record"}},"created_at":"2026-07-05T12:05:11.831560+00:00","updated_at":"2026-07-05T12:05:11.831560+00:00"}