{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:GJBFPRUP464PYUNCKV3UZYQJYV","short_pith_number":"pith:GJBFPRUP","schema_version":"1.0","canonical_sha256":"324257c68fe7b8fc51a255774ce209c54ba6ddc778aee5b7ed97ff847dc0bb52","source":{"kind":"arxiv","id":"1802.00350","version":3},"attestation_state":"computed","paper":{"title":"An $L^2$-identity and pinned distance problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2018-02-01T15:36:15Z","abstract_excerpt":"Let $\\mu$ be a Frostman measure on $E\\subset\\mathbb{R}^d$. The spherical average decay $$\\int_{S^{d-1}}|\\widehat{\\mu}(r\\omega)|^2\\,d\\omega\\lesssim r^{-\\beta} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly.\n  The idea is to r"},"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":"1802.00350","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-02-01T15:36:15Z","cross_cats_sorted":["math.AP","math.CO","math.MG"],"title_canon_sha256":"5820b63a13e4504a36022d54a9bd9bee457ec647be0198245aa28dc6ebbd891c","abstract_canon_sha256":"ef7959377e0e84915f011e1dcafe9b0b3a1b729645bab2e66ce7d22f517270a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:09.054440Z","signature_b64":"dUXzEdjqiXOjasQW5RiAL3V64caPbXZyVh6M3M2euXqHhcfDKh5Vp4r3jS19WJmk39gz9cOg8BA+JOcXWpJnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"324257c68fe7b8fc51a255774ce209c54ba6ddc778aee5b7ed97ff847dc0bb52","last_reissued_at":"2026-05-17T23:40:09.053718Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:09.053718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An $L^2$-identity and pinned distance problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2018-02-01T15:36:15Z","abstract_excerpt":"Let $\\mu$ be a Frostman measure on $E\\subset\\mathbb{R}^d$. The spherical average decay $$\\int_{S^{d-1}}|\\widehat{\\mu}(r\\omega)|^2\\,d\\omega\\lesssim r^{-\\beta} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly.\n  The idea is to r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00350","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":""},"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":"1802.00350","created_at":"2026-05-17T23:40:09.053837+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.00350v3","created_at":"2026-05-17T23:40:09.053837+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00350","created_at":"2026-05-17T23:40:09.053837+00:00"},{"alias_kind":"pith_short_12","alias_value":"GJBFPRUP464P","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_16","alias_value":"GJBFPRUP464PYUNC","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_8","alias_value":"GJBFPRUP","created_at":"2026-05-18T12:32:25.280505+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/GJBFPRUP464PYUNCKV3UZYQJYV","json":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV.json","graph_json":"https://pith.science/api/pith-number/GJBFPRUP464PYUNCKV3UZYQJYV/graph.json","events_json":"https://pith.science/api/pith-number/GJBFPRUP464PYUNCKV3UZYQJYV/events.json","paper":"https://pith.science/paper/GJBFPRUP"},"agent_actions":{"view_html":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV","download_json":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV.json","view_paper":"https://pith.science/paper/GJBFPRUP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.00350&json=true","fetch_graph":"https://pith.science/api/pith-number/GJBFPRUP464PYUNCKV3UZYQJYV/graph.json","fetch_events":"https://pith.science/api/pith-number/GJBFPRUP464PYUNCKV3UZYQJYV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV/action/storage_attestation","attest_author":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV/action/author_attestation","sign_citation":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV/action/citation_signature","submit_replication":"https://pith.science/pith/GJBFPRUP464PYUNCKV3UZYQJYV/action/replication_record"}},"created_at":"2026-05-17T23:40:09.053837+00:00","updated_at":"2026-05-17T23:40:09.053837+00:00"}