{"paper":{"title":"Distinct and repeated distances on a surface and incidences between points and spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Micha Sharir, Noam Solomon","submitted_at":"2016-04-06T06:20:11Z","abstract_excerpt":"In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\\mathbb R^3$ is at least $\\Omega\\left(n^{7/9}/{\\rm polylog} \\,n\\right)$. This bound is significantly larger than the conjectured bound $\\Omega(n^{2/3})$ for general point sets in $\\mathbb R^3$.\n  We also show that the number of unit distances determined by $n$ points on a surface $V$, as above, is $O(n^{4/3})$, a bound that matches the best known planar bound, and is worst-case tight in 3-space. This is in sharp contrast "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01502","kind":"arxiv","version":1},"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"}