{
  "abstract": "We give counterexamples to two optimization problems in dimensions eight and nine. 1. The Cartesian-product problem posed by Cassels for critical determinants and later formulated by Zong for lattice packings and for packings allowing translations but not rotations: whether the corresponding product inequalities are always equalities. The counterexample also disproves the natural product formula for unrestricted congruent packings. 2. A question raised by Sarnak and formulated as a conjecture in Chiu: whether, among unit-volume flat tori, height is minimized by a lattice maximizing the length of its shortest nonzero vector.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.11695",
  "author_total": 1,
  "authors": [
    {
      "name": "Nihar Gargava"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c50t7rwvgktgs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.11695",
  "fulltext_url": null,
  "id": "c50t7rwvgktgs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.11695",
  "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.MG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7168290244",
  "pdf_url": "https://arxiv.org/pdf/2607.11695",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c50t7rwvgktgs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2607.11695",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.11695"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On two counterexamples in the geometry of numbers",
  "title_text": "On two counterexamples in the geometry of numbers",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/c50t7rwvgktgs",
  "venue": "arXiv",
  "versions": []
}