{
  "abstract": "We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $φ(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\\#\\left\\{1\\leq x< p/2:\\left\\{{φ(x)}/{p}\\right\\}>\\frac12\\right\\} =\\frac{p}{4}+O_φ(\\sqrt p\\log^2 p). $ We then show that the error term can be improved to $O_φ(\\sqrt p\\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials $φ(x)=x^m$, we further obtain the bound $O_m(\\sqrt p\\log\\log p)$ under the Generalized Riemann Hypothesis. Finally, in the case $m=2$, we prove an unconditional matching lower bound, showing that the factor $\\log\\log p$ is best possible in this setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.21259",
  "author_total": 3,
  "authors": [
    {
      "name": "Xuejun Guo"
    },
    {
      "name": "Chen Lin"
    },
    {
      "name": "Zhefeng Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f81yxhjv3424q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.21259",
  "fulltext_url": null,
  "id": "f81yxhjv3424q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.21259",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.21259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f81yxhjv3424q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.21259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.21259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Fractional Parts of Polynomials Modulo $p$",
  "title_text": "On the Fractional Parts of Polynomials Modulo p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/f81yxhjv3424q",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.ffa.2026.102923",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}