{"paper":{"title":"Four-variable expanders over the prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Claudiu Valculescu, Doowon Koh, Hossein Nassajian Mojarrad, Thang Pham","submitted_at":"2017-05-11T16:04:23Z","abstract_excerpt":"Let $\\mathbb{F}_p$ be a prime field of order $p>2$, and $A$ be a set in $\\mathbb{F}_p$ with very small size in terms of $p$. In this note, we show that the number of distinct cubic distances determined by points in $A\\times A$ satisfies \\[|(A-A)^3+(A-A)^3|\\gg |A|^{8/7},\\] which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables.\n  We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that \\[\\max \\left\\lbrace |A+A|, |f(A, A)|\\right\\rbrace\\gg |A|^{6/5},\\] where $f(x,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04255","kind":"arxiv","version":6},"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"}