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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","authors_text":"Dung The Tran, Steven Senger","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-01-12T00:05:47Z","title":"A sharp point-sphere incidence bound for $(u, s)$-Salem sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.07105","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-16T15:58:09.643285Z","id":"db758d47-d17b-49e7-9914-d6fd78fd5666","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"","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}.","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."}},"verdict_id":"db758d47-d17b-49e7-9914-d6fd78fd5666"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:88b438ee32b56f8d46bf710a410e81779926074d09a6e23e646bddb94bb25389","target":"record","created_at":"2026-06-09T01:05:12Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e2697dfb259f785fed7f6e06ba54c3bf3ff8d908eca8dc5c0434f3b7338fc83f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-01-12T00:05:47Z","title_canon_sha256":"15d71f4b7c14ff779e6e3f4c1844dc903e777b458256b0dfd33dad47e7ce87b4"},"schema_version":"1.0","source":{"id":"2601.07105","kind":"arxiv","version":4}},"canonical_sha256":"adb61f589058046d9ac4ff014bb70c4325bea4c95dad1da51a78fd456f6d4640","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"adb61f589058046d9ac4ff014bb70c4325bea4c95dad1da51a78fd456f6d4640","first_computed_at":"2026-06-09T01:05:12.583260Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:12.583260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0ByMaFcJs+sHhRPljWKdLsOuwYpibRsnAMk8v9jjpl1OU3bOlyAaPdOpGjPoTSiufsYptRE2jwHj0iCKwJAdCQ==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:12.583659Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.07105","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88b438ee32b56f8d46bf710a410e81779926074d09a6e23e646bddb94bb25389","sha256:0afc3be27943c06f929450f1bb0d3b2bebffc9969d276e5102f9c359a54ddcf0"],"state_sha256":"9d5cacf0eaff29f00fad1fae49091b821f0d1f12da5629d25a4cd0d4c63323d7"}