{"paper":{"title":"Spectral Bounds for Antipodal Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Samuel Korsky","submitted_at":"2026-03-11T02:06:10Z","abstract_excerpt":"Suppose $\\left\\{x_1, \\dots, x_n\\right\\} \\subset \\mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\\leq 1$, meaning $|x_i - x_j| \\leq 1$ for all $1 \\leq i,j \\leq n$. We show that the ratio of the number of ``neighbors'' (ordered pairs of points with distance $\\leq \\varepsilon$) to the number of ``antipodes'' (ordered pairs of points with distance $\\geq 1 - \\varepsilon$) is $\\gtrsim\\varepsilon^{1/2 + o(1)}$, attaining the conjectured correct asymptotic within a polylog factor and improving the $\\gtrsim\\varepsilon^{3/4+o(1)}$ bound of Steinerberger (2025). In dimensions $d\\ge3$ we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.10334","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.10334/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}