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Then the union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a slightly weakened version of a result of Bukh."},"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":"1112.3468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-15T10:17:53Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"512bea8410e542059ebde42f41e7795887314670b9b2c0abd03364eb79413097","abstract_canon_sha256":"1b2ddbe93da74a36658066d6dfb190a971befc5c96d2f3517aa08efc0feb6698"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:18.516452Z","signature_b64":"tKc+KjnnF5JPDktI6MyjzrlarIPMZW/DeMJBw8zWU6ZL+4muzCJayZpMjZZRzbufYXjiEcINKRdyq+QAP+hxAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1136de4e7b1a90e22f705ee2716752f5a56597d6870c7155e2aa3c4e34af9eed","last_reissued_at":"2026-05-18T04:06:18.515731Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:18.515731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Contractions and expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ben Green, Emmanuel Breuillard","submitted_at":"2011-12-15T10:17:53Z","abstract_excerpt":"Let A be a finite set of reals and let K >= 1 be a real number. 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