{"paper":{"title":"Incidences and pairs of dot products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Lund","submitted_at":"2015-09-03T13:22:41Z","abstract_excerpt":"Let $\\mathbb{F}$ be a field, let $P \\subseteq \\mathbb{F}^d$ be a finite set of points, and let $\\alpha,\\beta \\in \\mathbb{F} \\setminus \\{0\\}$. We study the quantity \\[|\\Pi_{\\alpha, \\beta}| = \\{(p,q,r) \\in P \\times P \\times P \\mid p \\cdot q = \\alpha, p \\cdot r = \\beta \\}.\\]\n  We observe a connection between the question of placing an upper bound on $|\\Pi_{\\alpha,\\beta}|$ and a well-studied question on the number of incidences betwen points and hyperplanes, and use this connection to prove new and strengthened upper bounds on $|\\Pi_{\\alpha,\\beta}|$ in a variety of settings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01072","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"}