{"paper":{"title":"On distinct perpendicular bisectors and pinned distances in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ben Lund, Brandon Hanson, Oliver Roche-Newton","submitted_at":"2014-12-04T10:38:15Z","abstract_excerpt":"Given a set of points $P \\subset \\mathbb F_q^2$ such that $|P|\\geq q^{3/2}$ it is established that $|P|$ determines $\\Omega(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \\geq q^{4/3}$, then for a positive proportion of points $a \\in P$, we have $$|\\{\\| a- b\\|: b \\in P\\}|=\\Omega(q),$$ where $\\|a- b\\|$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1611","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":""},"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"}