{
  "abstract": "Generalizing a recent theorem by Higashitani et al. (Comb. Theory 2025), it is shown that a convex body $K$ is uniquely described up to integer translation by the function $\\mathrm{t}_K(v) = |(K-v)\\cap\\mathbb{Z}^n|$, $v\\in\\mathbb{R}^n$. For a lattice polytope $P$ the function $\\mathrm{t}_P$ is decomposed into homogeneous components $\\mathrm{t}_{P,i}$, similar to the Ehrhart polynomial. It is shown that the functions $\\mathrm{t}_{P,i}$ are simultaneous eigenfunctions of a family of linear operators introduced by Kubert and Lang in (Math. Z. 1976) and that, conversely, the linear spaces of piecewise constant eigenfunctions are generated by the respective coefficients $\\mathrm{t}_{P,i}$. A structure theorem for the simultaneous eigenspace corresponding to the $(n-1)$st coefficient in the decomposition of $\\mathrm{t}_P$ is obtained. This leads to an explicit description of the set of functions $\\mathrm{t}_P$, where $P$ ranges over all lattice polygons, as well as to inequalities for the number of lattice points in dilated and translated lattice polytopes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26523",
  "author_total": 2,
  "authors": [
    {
      "name": "Ansgar Freyer"
    },
    {
      "name": "Georg C. Hofstätter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7skmsbj2vsyg1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26523",
  "fulltext_url": null,
  "id": "7skmsbj2vsyg1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26523",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223015153",
  "pdf_url": "https://arxiv.org/pdf/2610.26523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7skmsbj2vsyg1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Counting Lattice Points in Translations: Uniqueness and Existence Results",
  "title_text": "Counting Lattice Points in Translations: Uniqueness and Existence Results",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7skmsbj2vsyg1",
  "venue": "arXiv",
  "versions": []
}