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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.8064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-29T10:44:15Z","cross_cats_sorted":[],"title_canon_sha256":"528713c4c013b3311437ba0095949b08aaa9ee2acfbeecb546774aa456d14fe9","abstract_canon_sha256":"96f5da7b9bb629651a9af46e0488ae028c2247cd6122110a8e4f9e9ac3808e1c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:33.742672Z","signature_b64":"XgwANACQ2OPOeV9In5jNBaRqduLWad85oCcI596a8X4d394v9ecCkLeRVBJrwbwiPV2ilhDNKdmuflo1DFlgBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e755f750874e8dc297846e2bbec1b853a57edac3fd873b5663c87395553bbe42","last_reissued_at":"2026-05-18T02:41:33.742299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:33.742299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $A$). 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