{"paper":{"title":"Cauchy-Davenport type inequalities, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Salvatore Tringali","submitted_at":"2016-04-07T19:46:19Z","abstract_excerpt":"Let $\\mathbb G = (G, +)$ be a group (either abelian or not). Given $X, Y \\subseteq G$, we denote by $\\langle Y \\rangle$ the subsemigroup of $\\mathbb G$ generated by $Y$, and we set $$\\gamma(Y) := \\sup_{y_0 \\in Y} \\inf_{y_0 \\ne y \\in Y} {\\rm ord}(y - y_0)$$ if $|Y| \\ge 2$ and $\\gamma(Y) := |Y|$ otherwise. We prove that if $\\langle Y \\rangle$ is commutative, $Y$ is non-empty, and $X+2Y \\neq X + Y + y$ for some $y \\in Y$, then $$ |X+Y| \\ge |X|+\\min(\\gamma(Y), |Y| - 1). $$ Actually, this is obtained from a more general result, which improves on previous work of the author on sumsets in cancellativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02136","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"}