{"paper":{"title":"Bisector energy and pinned distances in positive characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan Murphy, Misha Rudnev, Sophie Stevens","submitted_at":"2019-08-13T12:57:09Z","abstract_excerpt":"We prove a new lower bound for the number of pinned distances over finite fields: if $A$ is a sufficiently small subset of $\\mathbb{F}_q^2$, then there is an element in $A$ that determines $\\gg |A|^{2/3}$ distinct distances to other elements of $A$. Combined with results for large subsets $A\\subseteq\\mathbb{F}_q^2$, this improves all previously known lower bounds on distinct distances over finite fields.\n  In fact, we obtain an upper bound for the number of isosceles triangles determined by $A$. For that we use the concept of bisector energy. It turns out that the latter can be expressed as a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04618","kind":"arxiv","version":3},"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/1908.04618/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"}