{
  "abstract": "The three gap theorem states that the points of the Kronecker sequence $α,2α,\\ldots,Nα$, considered modulo one, divide the circle into intervals of at most three distinct lengths. In a two-dimensional nearest-neighbour analogue, Haynes and Marklof proved that the Kronecker sequence modulo an arbitrary unimodular lattice determines at most five distinct nearest-neighbour distances in the Euclidean norm, and that this bound is sharp. Dettmann subsequently constructed examples attaining five distinct distances for every $\\ell_p$-norm, $1\\leq p\\leq\\infty$. We prove the corresponding upper bound for every norm on $\\mathbb R^2$: for every full-rank lattice $L$, every $α\\in\\mathbb R^2$, and every $N\\in\\mathbb N$, the number of distinct nearest-neighbour distances is at most five. For every norm, this bound is attained for some $L$ and $α$. For strictly convex norms, the proof extends the lattice-theoretic argument of Haynes and Marklof by replacing the Euclidean angular estimates with a cone lemma based on a proper Brass angular measure. The result for arbitrary norms is then obtained by a strictly convex perturbation and a limiting argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12405",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikita A. Mironov"
    },
    {
      "name": "Oleg R. Musin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59br9qapp8348.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12405",
  "fulltext_url": null,
  "id": "59br9qapp8348",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12405",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7203482875",
  "pdf_url": "https://arxiv.org/pdf/2608.12405",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59br9qapp8348.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2608.12405",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12405"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Five Distance Theorem For An Arbitrary Norm",
  "title_text": "The Five Distance Theorem For An Arbitrary Norm",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/59br9qapp8348",
  "venue": "arXiv",
  "versions": []
}