{
  "abstract": "For every $n\\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\\cdot2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\\ge3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\\ge4$. This number is optimal when $r=2^k+2$, $k\\ge1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26723",
  "author_total": 1,
  "authors": [
    {
      "name": "Florian Frick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5k6030c5ba7hs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26723",
  "fulltext_url": null,
  "id": "5k6030c5ba7hs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26723",
  "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.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214072144",
  "pdf_url": "https://arxiv.org/pdf/2609.26723",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5k6030c5ba7hs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26723",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26723"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Counterexamples to the Ramos conjecture for two hyperplanes",
  "title_text": "Counterexamples to the Ramos conjecture for two hyperplanes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/5k6030c5ba7hs",
  "venue": "arXiv",
  "versions": []
}