{"paper":{"title":"A sharp point-sphere incidence bound for $(u, s)$-Salem sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For (4,s)-Salem point sets P in F_q^d with |P| much smaller than q to the power d over 4s, the deviation of point-sphere incidences from the average is bounded by q to the d/4 times |P| to the 1-s times |S| to the 3/4.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dung The Tran, Steven Senger","submitted_at":"2026-01-12T00:05:47Z","abstract_excerpt":"We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \\((4,s)\\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if \\(P\\subset \\mathbb{F}_q^d\\) is a \\((4,s)\\)-Salem set with \\(s\\in \\big( \\frac{1}{4}, \\frac{1}{2} \\big]\\) and \\(|P|\\ll q^{ \\frac{d}{4s}}\\), then for any finite family \\(S\\) of spheres in \\(\\mathbb{F}_q^d\\), \\[ \\bigg| I(P,S)-\\frac{|P||S| }{q} \\bigg| \\ll q^{\\frac{d}{4}}\\,|P|^{1-s}\\,|S|^{\\frac{3}{4}}. \\] This estimate improves the classical point"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"If P subset F_q^d is a (4,s)-Salem set with s in (1/4, 1/2] and |P| << q^{d/(4s)}, then for any finite family S of spheres, |I(P,S) - |P||S|/q| << q^{d/4} |P|^{1-s} |S|^{3/4}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The point set P satisfies the (4,s)-Salem condition quantifying its fourth-order additive energy, together with the size restriction |P| << q^{d/(4s)} that enables the lifting argument to succeed.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For (4,s)-Salem point sets P in F_q^d with |P| much smaller than q to the power d over 4s, the deviation of point-sphere incidences from the average is bounded by q to the d/4 times |P| to the 1-s times |S| to the 3/4.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"8e7852a808d2e38860eb0c889c979fe067e55f9147f9d34c75ee129810839278"},"source":{"id":"2601.07105","kind":"arxiv","version":4},"verdict":{"id":"db758d47-d17b-49e7-9914-d6fd78fd5666","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T15:58:09.643285Z","strongest_claim":"If P subset F_q^d is a (4,s)-Salem set with s in (1/4, 1/2] and |P| << q^{d/(4s)}, then for any finite family S of spheres, |I(P,S) - |P||S|/q| << q^{d/4} |P|^{1-s} |S|^{3/4}.","one_line_summary":"For (4,s)-Salem point sets P in F_q^d with |P| much smaller than q to the power d over 4s, the deviation of point-sphere incidences from the average is bounded by q to the d/4 times |P| to the 1-s times |S| to the 3/4.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The point set P satisfies the (4,s)-Salem condition quantifying its fourth-order additive energy, together with the size restriction |P| << q^{d/(4s)} that enables the lifting argument to succeed.","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.07105/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":2,"snapshot_sha256":"865eb3c835734c069c7d69e65874dacb1870483010b5fcaec670d06757d52beb"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}