{
  "annotation_id": "f8c49731-dcf6-43e6-87e6-53deecd59820",
  "date_created": "2026-02-17T05:53:11.280000Z",
  "date_modified": "2026-02-17T05:53:11.280000Z",
  "file_hash": "f85fe0916b497f29fcdf34bbd0e5ac3d29608b3d0c4e0dc1a843eca67d0f8b1e",
  "private": false,
  "record": {
    "abstract": "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-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the \\((4,s)\\)-Salem property. As applications, we derive refined bounds for unit distances, dot-product configurations, and sum-product type phenomena, and we extend the method to \\((u,s)\\)-Salem sets for even moments \\(u\\ge4\\).",
    "arxiv_id": "2601.07105",
    "authors": [
      "Steven Senger",
      "Dung The Tran"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "A sharp point-sphere incidence bound for $(u, s)$-Salem sets",
    "url": "https://arxiv.org/abs/2601.07105",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "3fcd9dec-d6e6-4fda-ac1b-d459e6a0cbd2",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}