{
  "abstract": "Let $X=\\{x_1,\\dots,x_n\\}\\subset \\mathbb{R}^2$ be a finite set of points of diameter at most $1$. It is natural to expect that if many pairs $(x_i,x_j)$ lie at distance close to $1$ from each other, then some clustering phenomenon must occur, implying that a significant number of these pairs are also very close to each other. %For $0<\\varepsilon<1$, we call a pair $(x_i,x_j)$ $\\varepsilon$-antipodal if $\\|x_i-x_j\\|\\ge 1-\\varepsilon$ and $\\varepsilon$-neighboring if $\\|x_i-x_j\\|\\le \\varepsilon$. We prove that there exists a universal constant $c>0$ such that for all $0<\\varepsilon<1$, whenever $n$ is large enough, we have: \\[ \\big|\\{(i,j):\\|x_i-x_j\\|\\le \\varepsilon\\}\\big| \\geq c\\cdot \\varepsilon^{1/2}\\cdot \\big|\\{(i,j):\\|x_i-x_j\\|\\geq 1-\\varepsilon\\}\\big|. \\] This confirms a recent conjecture of Steinerberger, who asked whether the $\\varepsilon^{1/2}$ ratio is the best possible. We also study a two-parameter version of Steinerberger's question by considering the number of pairs at distance at most $\\varepsilon_1$ and at distance at least $1-\\varepsilon_2$. We show that in this case the optimal ratio is $\\varepsilon_1^2\\cdot\\varepsilon_2^{-3/2}$ when $\\varepsilon_1\\leq\\varepsilon_2$, and $\\varepsilon_1^{1/2}$ otherwise. The proof proceeds by introducing an auxiliary graph associated with the set $X$ and reducing the problem to bounding the largest eigenvalue of its adjacency matrix. Our main result is the outcome of human--AI interactions using ChatGPT 5.4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02605",
  "author_total": 2,
  "authors": [
    {
      "name": "Gábor Damásdi"
    },
    {
      "name": "Laurentiu Ploscaru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/985xeghg56a3a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02605",
  "fulltext_url": null,
  "id": "985xeghg56a3a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02605",
  "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",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7172525989",
  "pdf_url": "https://arxiv.org/pdf/2608.02605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/985xeghg56a3a.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2608.02605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Many Antipodal Pairs Force Many Neighboring Pairs",
  "title_text": "Many Antipodal Pairs Force Many Neighboring Pairs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/985xeghg56a3a",
  "venue": "arXiv",
  "versions": []
}