{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:O5BX562ZIMEEDBP7O5LHEZPRZW","short_pith_number":"pith:O5BX562Z","schema_version":"1.0","canonical_sha256":"77437efb5943084185ff77567265f1cdb50fa741ff06c7f68a93fc95c9b84363","source":{"kind":"arxiv","id":"2510.26364","version":3},"attestation_state":"computed","paper":{"title":"Additive structures imply more distances in $\\mathbb{F}_q^d$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"Daewoong Cheong, Doowon Koh, Dung The Tran, Gennian Ge, Tao Zhang, Thang Pham","submitted_at":"2025-10-30T11:11:00Z","abstract_excerpt":"For a set $E \\subseteq \\mathbb{F}_q^d$, the distance set is defined as $\\Delta(E) := \\{\\|\\mathbf{x} - \\mathbf{y}\\| : \\mathbf{x}, \\mathbf{y} \\in E\\}$, where $\\|\\cdot\\|$ denotes the standard quadratic form. We investigate the Erd\\H{o}s--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Jonathan M. Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\\|\\widehat{E}\\|_4$ and the fourth additive energy $\\Lambda_4(E)$, we prove that quantitative gains in $\\Lambda_4(E)$ force the existence of many distances.\n  In particular, "},"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":"2510.26364","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-10-30T11:11:00Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"7b912bebc09002394cdd465433fe1a0bf62c4f70ed8c2b5232b602154147f544","abstract_canon_sha256":"29dc504d8ed071bb9dc93d5efac8a09db2ecc219392b700793ef1c65efe15e4a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T02:04:44.084770Z","signature_b64":"AMTHj7EPp7VP6sRAkjWKsDdPf5dccaRfqO+SV2tZxewHocWvlU5rJHYoo1ulTqT0YRgsKsU8kbF73jw/neOLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77437efb5943084185ff77567265f1cdb50fa741ff06c7f68a93fc95c9b84363","last_reissued_at":"2026-05-28T02:04:44.084086Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T02:04:44.084086Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additive structures imply more distances in $\\mathbb{F}_q^d$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"Daewoong Cheong, Doowon Koh, Dung The Tran, Gennian Ge, Tao Zhang, Thang Pham","submitted_at":"2025-10-30T11:11:00Z","abstract_excerpt":"For a set $E \\subseteq \\mathbb{F}_q^d$, the distance set is defined as $\\Delta(E) := \\{\\|\\mathbf{x} - \\mathbf{y}\\| : \\mathbf{x}, \\mathbf{y} \\in E\\}$, where $\\|\\cdot\\|$ denotes the standard quadratic form. We investigate the Erd\\H{o}s--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Jonathan M. Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\\|\\widehat{E}\\|_4$ and the fourth additive energy $\\Lambda_4(E)$, we prove that quantitative gains in $\\Lambda_4(E)$ force the existence of many distances.\n  In particular, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.26364","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/2510.26364/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":"2510.26364","created_at":"2026-05-28T02:04:44.084180+00:00"},{"alias_kind":"arxiv_version","alias_value":"2510.26364v3","created_at":"2026-05-28T02:04:44.084180+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.26364","created_at":"2026-05-28T02:04:44.084180+00:00"},{"alias_kind":"pith_short_12","alias_value":"O5BX562ZIMEE","created_at":"2026-05-28T02:04:44.084180+00:00"},{"alias_kind":"pith_short_16","alias_value":"O5BX562ZIMEEDBP7","created_at":"2026-05-28T02:04:44.084180+00:00"},{"alias_kind":"pith_short_8","alias_value":"O5BX562Z","created_at":"2026-05-28T02:04:44.084180+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2601.07105","citing_title":"A sharp point-sphere incidence bound for $(u, s)$-Salem sets","ref_index":3,"is_internal_anchor":true},{"citing_arxiv_id":"2604.19486","citing_title":"On Fourier decay and the distance set problem","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW","json":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW.json","graph_json":"https://pith.science/api/pith-number/O5BX562ZIMEEDBP7O5LHEZPRZW/graph.json","events_json":"https://pith.science/api/pith-number/O5BX562ZIMEEDBP7O5LHEZPRZW/events.json","paper":"https://pith.science/paper/O5BX562Z"},"agent_actions":{"view_html":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW","download_json":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW.json","view_paper":"https://pith.science/paper/O5BX562Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2510.26364&json=true","fetch_graph":"https://pith.science/api/pith-number/O5BX562ZIMEEDBP7O5LHEZPRZW/graph.json","fetch_events":"https://pith.science/api/pith-number/O5BX562ZIMEEDBP7O5LHEZPRZW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW/action/storage_attestation","attest_author":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW/action/author_attestation","sign_citation":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW/action/citation_signature","submit_replication":"https://pith.science/pith/O5BX562ZIMEEDBP7O5LHEZPRZW/action/replication_record"}},"created_at":"2026-05-28T02:04:44.084180+00:00","updated_at":"2026-05-28T02:04:44.084180+00:00"}