{
  "abstract": "Let $\\mathcal{N}(k,n)$ be the largest size of a family of affine $k$-planes in $\\R^n$ at pairwise separation distance one. We develop a unified treatment of upper bounds, dimension reduction, and product constructions. Orthogonal-projection coordinates give a rank-stratified distance identity; a Veronese lifting converts its polynomial consequences into simultaneous total isotropy. Connected Nash decomposition and an analytic diagonal argument replace the linear-independence step in Höfer's three-dimensional method. An explicit cylindrical decomposition estimate yields $\\mathcal{N}(k,n)\\leqslant 2^{\\,2^{\\,3(k+1)(n-k)}}$. For configurations whose direction sums have dimension at most $k+1$, we prove the exact reduction $\\mathcal{N}_{\\leqslant k+1}(k,n)=\\mathcal{N}(1,n-k+1)$. Every codimension-two equidistant configuration splits off a common linear factor, giving $\\mathcal{N}(n-2,n)=\\mathcal{N}(1,3)$. We also determine the sharp bound $2(n-k)$ when all directions lie in a fixed $(k+1)$-dimensional subspace. A transverse product construction gives multiplicative inequalities and makes $S(c)=\\mathcal{N}(c-1,2c-1)$ supermultiplicative. In particular, $S(c)\\geqslant 2^{c\\bmod 2}7^{\\lfloor c/2\\rfloor}$. The structural results and constructions do not depend on the exact value of $\\mathcal{N}(1,3)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37421",
  "author_total": 1,
  "authors": [
    {
      "name": "Jia Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3n03ttcrk6cfx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37421",
  "fulltext_url": null,
  "id": "3n03ttcrk6cfx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37421",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7224126345",
  "pdf_url": "https://arxiv.org/pdf/2610.37421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3n03ttcrk6cfx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Equidistant Affine Subspaces: Explicit Bounds, Codimension-Two Reduction, and Transverse Products",
  "title_text": "Equidistant Affine Subspaces: Explicit Bounds, Codimension-Two Reduction, and Transverse Products",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3n03ttcrk6cfx",
  "venue": "arXiv",
  "versions": []
}