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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"},"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":true},"canonical_record":{"source":{"id":"2601.07105","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-01-12T00:05:47Z","cross_cats_sorted":[],"title_canon_sha256":"15d71f4b7c14ff779e6e3f4c1844dc903e777b458256b0dfd33dad47e7ce87b4","abstract_canon_sha256":"e2697dfb259f785fed7f6e06ba54c3bf3ff8d908eca8dc5c0434f3b7338fc83f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:12.583659Z","signature_b64":"0ByMaFcJs+sHhRPljWKdLsOuwYpibRsnAMk8v9jjpl1OU3bOlyAaPdOpGjPoTSiufsYptRE2jwHj0iCKwJAdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"adb61f589058046d9ac4ff014bb70c4325bea4c95dad1da51a78fd456f6d4640","last_reissued_at":"2026-06-09T01:05:12.583260Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:12.583260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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