{"paper":{"title":"A note on the combinatorial derivation of non-small sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joshua Erde","submitted_at":"2014-09-29T10:44:15Z","abstract_excerpt":"Given an infinite group $G$ and a subset $A$ of $G$ we let $\\Delta(A) = \\{g \\in G \\,:\\, |gA \\cap A| =\\infty\\}$ (this is sometimes called the \\emph{combinatorial derivation} of $A$). A subset $A$ of $G$ is called: \\emph{large} if there exists a finite subset $F$ of $G$ such that $FA=G$; \\emph{$\\Delta$-large} if $\\Delta(A)$ is large and \\emph{small} if for every large subset $L$ of $G$, $(G \\setminus A) \\cap L$ is large. In this note we show that every non-small set is $\\Delta$-large, answering a question of Protasov."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8064","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"}