{"paper":{"title":"Convexity and a sum-product type estimate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Liangpan Li, Oliver Roche-Newton","submitted_at":"2011-11-22T11:19:34Z","abstract_excerpt":"In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\\cite{ENR}). In particular, we show that for any finite set $A\\subset{\\mathbb{R}}$ and any strictly convex or concave function $f$, \\[|A+f(A)|\\gg{\\frac{|A|^{24/19}}{(\\log|A|)^{2/19}}}\\] and \\[\\max\\{|A-A|,\\ |f(A)+f(A)|\\}\\gg{\\frac{|A|^{14/11}}{(\\log|A|)^{2/11}}}.\\] For the latter of these inequalities, we go on to consider the consequences for a sum-product type problem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5159","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"}