{
  "abstract": "We develop two transition principles for lower-bounding value sets generated by structured sequences over prime fields. A reciprocal-affine family with $M$ internal transitions and bounded quotient multiplicity has image size $\\gg \\min{M,p}^{8/15}$. This recovers the factorial-residue bound and yields the same exponent for arithmetic Pochhammer products, Gaussian $q$-factorials, derangement numbers, and the numbers of ordered subsets. A second theorem treats nonzero sequences whose consecutive ratios evolve under a nondegenerate M\"obius transformation: their value sets have size $\\gg \\min{M,p}^{1/2+η}$ for an absolute constant $η>0$. As consequences, fixed rows of Pascal's triangle and the initial half-blocks of the Catalan and central binomial sequences exceed the square-root scale. The proofs combine transition quotients with, respectively, Cartesian-product point-line incidence geometry and Bourgain's expansion-based incidence theorem in $\\mathrm{SL}_2(\\mathbb{F}_p)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05652",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiyu Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ck60zfk60c6c7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05652",
  "fulltext_url": null,
  "id": "ck60zfk60c6c7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05652",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7211996999",
  "pdf_url": "https://arxiv.org/pdf/2609.05652",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ck60zfk60c6c7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05652",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05652"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem",
  "title_text": "Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/ck60zfk60c6c7",
  "venue": "arXiv",
  "versions": []
}