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The framework reduces the natural \\(q^m\\)-scale positive-proportion theorem to a uniform \\(L^1\\) anti-concentration statement for positive convex combinations of classical Kloosterman sums. Assuming this Kloosterman anti-concentration conjecture, we prove that for every \\(0<\\alpha<\\frac{1}{2}\\) there is a constant \\(C_{m,\\alpha}\\) such that \\[\n  \\min\\{|E|,|F| \\} \\ge C_{m,\\alpha}q^m\n  "},"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":"2607.05926","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T07:30:38Z","cross_cats_sorted":[],"title_canon_sha256":"5a02cf7b7ab21c2609f032e013f2d490b232dd529626263e36f89a53213822a5","abstract_canon_sha256":"25a9976380cc4741a34eae357ce902a718afb1b88929d7c5a3abe59dd2cdcc45"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:18:50.484882Z","signature_b64":"0J+4CwSGWBbjMQIQBitMRvkWNsio8zxiKYfWnslYet9IIZTif/2f7yG4AGCWzg96wGMudHbqXuxLqNbQSF1mDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6e9f86ef8d47b9b9a076234c6a4a75db4b2a1f1c32d0b667bea747c1d6d54e1","last_reissued_at":"2026-07-08T01:18:50.484406Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:18:50.484406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Erd\\H{o}s--Falconer distance conjecture from an analytic perspective","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dung The Tran, Le Quang Ham","submitted_at":"2026-07-07T07:30:38Z","abstract_excerpt":"Let \\(q\\) be an odd prime power and let \\(V=\\F_q^{2m}\\), equipped with \\(Q(x)=x_1^2+\\cdots+x_{2m}^2\\). 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Assuming this Kloosterman anti-concentration conjecture, we prove that for every \\(0<\\alpha<\\frac{1}{2}\\) there is a constant \\(C_{m,\\alpha}\\) such that \\[\n  \\min\\{|E|,|F| \\} \\ge C_{m,\\alpha}q^m\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05926","kind":"arxiv","version":1},"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/2607.05926/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":"2607.05926","created_at":"2026-07-08T01:18:50.484470+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.05926v1","created_at":"2026-07-08T01:18:50.484470+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05926","created_at":"2026-07-08T01:18:50.484470+00:00"},{"alias_kind":"pith_short_12","alias_value":"63U7Q3XY2R5Z","created_at":"2026-07-08T01:18:50.484470+00:00"},{"alias_kind":"pith_short_16","alias_value":"63U7Q3XY2R5ZXGQH","created_at":"2026-07-08T01:18:50.484470+00:00"},{"alias_kind":"pith_short_8","alias_value":"63U7Q3XY","created_at":"2026-07-08T01:18:50.484470+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/63U7Q3XY2R5ZXGQHMI2MNJFHLW","json":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW.json","graph_json":"https://pith.science/api/pith-number/63U7Q3XY2R5ZXGQHMI2MNJFHLW/graph.json","events_json":"https://pith.science/api/pith-number/63U7Q3XY2R5ZXGQHMI2MNJFHLW/events.json","paper":"https://pith.science/paper/63U7Q3XY"},"agent_actions":{"view_html":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW","download_json":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW.json","view_paper":"https://pith.science/paper/63U7Q3XY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.05926&json=true","fetch_graph":"https://pith.science/api/pith-number/63U7Q3XY2R5ZXGQHMI2MNJFHLW/graph.json","fetch_events":"https://pith.science/api/pith-number/63U7Q3XY2R5ZXGQHMI2MNJFHLW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW/action/storage_attestation","attest_author":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW/action/author_attestation","sign_citation":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW/action/citation_signature","submit_replication":"https://pith.science/pith/63U7Q3XY2R5ZXGQHMI2MNJFHLW/action/replication_record"}},"created_at":"2026-07-08T01:18:50.484470+00:00","updated_at":"2026-07-08T01:18:50.484470+00:00"}