{"paper":{"title":"A refined energy bound for perpendicular bisectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Lund","submitted_at":"2016-04-07T16:14:40Z","abstract_excerpt":"Let $\\mathcal{P}$ be a set of $n$ points in the Euclidean plane. We prove that, for any $\\epsilon > 0$, either a single line or circle contains $n/2$ points of $\\mathcal{P}$, or the number of distinct perpendicular bisectors determined by pairs of points in $\\mathcal{P}$ is $\\Omega(n^{52/35 - \\epsilon})$, where the constant implied by the $\\Omega$ notation depends on $\\mathcal{P}$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains $n/2$ points of $\\mathcal{P}$, or the number of distinct perpendicular bisectors is $\\Omega(n^2)$.\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02059","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}