{"paper":{"title":"Three-variable expanding polynomials and higher-dimensional distinct distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Frank de Zeeuw, Le Anh Vinh, Thang Pham","submitted_at":"2016-12-29T04:13:59Z","abstract_excerpt":"We determine which quadratic polynomials in three variables are expanders over an arbitrary field $\\mathbb{F}$. More precisely, we prove that for a quadratic polynomial $f\\in \\mathbb{F}[x,y,z]$, which is not of the form $g(h(x)+k(y)+l(z))$, we have $|f(A\\times B\\times C)|\\gg N^{3/2}$ for any sets $A,B,C\\subset \\mathbb{F}$ with $|A|=|B|=|C|=N$, with $N$ not too large compared to the characteristic of $\\mathbb{F}$. We give several applications. We use this result for $f=(x-y)^2+z$ to obtain new lower bounds on $|A+A^2|$ and $\\max\\{|A+A|,|A^2+A^2|\\}$, and to prove that a Cartesian product $A\\time"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09032","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"}