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This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^"},"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":"0902.3506","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-02-20T03:18:49Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"3dea3bf09c6fbd3bad16440711b6d85216bcdc235b3185d435c679ed28c0ec75","abstract_canon_sha256":"251956fc4e845e6df225820873f7496ed40f07dbdacd0053c7da6e23dc5df7d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:05.307680Z","signature_b64":"i3njidvgF3DtUW9QYjQRRTEeB2BtPkIUyJh/AhxvzYCoPcwGiiPnIOPJ1kPnEc8XxGpKd4b/QvGlvkCQuJfMDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd22f17ea2d9618816e5f5b4b714d29e132da403708206fc149c1d9436bfb331","last_reissued_at":"2026-05-18T04:41:05.307257Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:05.307257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums and Products of Distinct Sets and Distinct Elements in $\\mathbb{C}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Karsten Chipeniuk","submitted_at":"2009-02-20T03:18:49Z","abstract_excerpt":"Let $A$ and $B$ be finite subsets of $\\mathbb{C}$ such that $|B|=C|A|$. 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