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We prove that, if $C$ is a finite subset of $G$ containing the identity element such that $\\langle C\\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\\geq 7$, $|BC|\\geq |B| + |C| + 2$. Also, we prove that if $C$ is a finite subset containing the identity element of a torsion-free group $G$ such that $|C| = 3$ and $\\langle C\\rangle$ is not abelian, then for all subsets $B$ of $G$ with"},"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":"1808.08708","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-27T07:11:38Z","cross_cats_sorted":["math.NT","math.RA"],"title_canon_sha256":"b8189ff1e1d9a7b88a0a610ca1ac20311e95205ee4f135db230d2e66ec7f0ef4","abstract_canon_sha256":"bcbe021d4b039a60e27626fd185fd54a1b8e888889bfbfaa54b22c7895ef24a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:53.908514Z","signature_b64":"AVe5goNFPmALZOy7DkbEMthw2AitFXfh6fTjH0z2ntVYxazuBik+T5UkUZgFJ4a7uXi4iNfiDEHWdDBjbKOuCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d86200001b8aa6cb484790825dc2a71324297df7e273435b2ddf59ad0e3a08ab","last_reissued_at":"2026-05-17T23:54:53.907758Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:53.907758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cardinality of product sets in torsion-free groups and applications in group algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.RA"],"primary_cat":"math.GR","authors_text":"Alireza Abdollahi, Fatemeh Jafari","submitted_at":"2018-08-27T07:11:38Z","abstract_excerpt":"Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $x\\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. 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