{"paper":{"title":"A new bound on Erd\\H{o}s distinct distances problem in the plane over prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Chun-Yen Shen, Doowon Koh, Le Anh Vinh, Thang Pham","submitted_at":"2018-05-22T23:01:43Z","abstract_excerpt":"In this paper we obtain a new lower bound on the Erd\\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\\subset \\mathbb{F}_p^2$ with $|A|\\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\\Delta(A)| \\gg |A|^{\\frac{1}{2}+\\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\\Delta{(A)}|$ in the range $|A|\\le p^{1+\\frac{149}{4065}}$.\n  The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08900","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"}