{
  "abstract": "We prove, in every dimension, the upper bound conjectured by Betke, Henk and Wills for the lattice-point enumerator in terms of volume and successive minima. For every convex body $K\\subset R^n$ and full-rank lattice $Λ$, we establish $$ \\#(K\\capΛ)\\le\\frac{\\mathrm{vol}_n(K)}{\\detΛ}\\prod_{i=1}^n\\left(1+\\frac{iλ_i}{2}\\right), $$ where $λ_i$ are the successive minima of $(K-K)/2$ with respect to $Λ$. Equality holds precisely for lattice translates of positive integral dilates of unimodular simplices. We also prove $$ \\#(\\mathrm{int}K\\capΛ)\\ge\\frac{\\mathrm{vol}_n(K)}{\\detΛ} \\left(1-\\frac{n}{4}\\sum_{i=1}^{n-1}λ_i-\\frac{n}{2}λ_n\\right), $$ which implies the lower estimate with factor $1-\\frac{1}{2}\\sum_{i=1}^n iλ_i$. In dimension two we prove the lower bound with factor $(1-λ_1/2)(1-λ_2)$ when $λ_2\\le1$. We also observe that a conjectured product lower bound for origin-symmetric bodies fails in dimension three. The proofs combine geometric compression, lattice-cell counts in coordinate sections, and an elementary coefficient comparison.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41125",
  "author_total": 2,
  "authors": [
    {
      "name": "Jun Wang"
    },
    {
      "name": "Fei Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3a25e32vkefdb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41125",
  "fulltext_url": null,
  "id": "3a25e32vkefdb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41125",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41125",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3a25e32vkefdb.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41125",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41125"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Bounds for lattice points in terms of volume and successive minima",
  "title_text": "Bounds for lattice points in terms of volume and successive minima",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/3a25e32vkefdb",
  "venue": "arXiv",
  "versions": []
}