{"paper":{"title":"On the number of unit-area triangles spanned by convex grids in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilya D. Shkredov, Micha Sharir, Orit E. Raz","submitted_at":"2015-04-27T09:25:59Z","abstract_excerpt":"A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A, B\\subset\\mathbb R$, each of size $n^{1/2}$, the convex grid $A\\times B$ spans at most $O(n^{37/17}\\log^{2/17}n)$ unit-area triangles. This improves the best known upper bound $O(n^{31/14})$ recently obtained in \\cite{RS}. Our analysis also applies to more general families of sets $A$, $B$, known as sets of Szemer\\'edi--Trotter type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06989","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"}