{"paper":{"title":"Improved bounds for incidences between points and circles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Joshua Zahl, Micha Sharir","submitted_at":"2012-07-31T23:20:20Z","abstract_excerpt":"We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension $\\ge 2$, is $O*(m^{2/3}n^{2/3} + m^{6/11}n^{9/11}+m+n)$, where the $O*(\\cdot)$ notation hides sub-polynomial factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in R^3 without first improving it in the plane.\n  Nevertheless, we show that if the set of circles is required to be \"truly three-dimensional\" i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0053","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":""},"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"}