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For any $g \\geq h \\geq 2$, we prove that if $A \\subset \\{1,2, \\dots ,n \\}$ is a $C_h[g]$-set in $\\mathbb{Z}$, then $|A| \\leq (g-1)^{1/h} n^{1 - 1/h} + O(n^{1/2 - 1/2h})$. We show that for any integer $n \\geq 1$, there is a $C_3 [3]$-set $A \\subset \\{1,2, \\dots , n \\}$ with $|A| \\geq (4^{-2/3} + o(1)) n^{2/3}$. We als"},"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":"1311.2985","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2013-11-12T23:59:48Z","cross_cats_sorted":[],"title_canon_sha256":"3904ffeaf61c12139b07641ff57b61556fa053b493ea09ed29b72edc435e473e","abstract_canon_sha256":"bf33e78e23ecaef7b21b0a927f61160da01f9c925ed22e96885d3cfb528f1de6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:13.484934Z","signature_b64":"80qX3PRqGxEXnNzXEyqtEDeQ9fjr2sSbHi+ZnpWr1fcpxqo978AbUFceXN6H02ZDa2ytaKiakIzgj5lDb9dMCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cbe174e69fe739a925b64cda822cc777e359124dfb379f6bd887fec9a1edc15","last_reissued_at":"2026-05-18T03:07:13.484422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:13.484422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds for generalized Sidon sets","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Craig Timmons, Rafael Tesoro, Xing Peng","submitted_at":"2013-11-12T23:59:48Z","abstract_excerpt":"Let $\\Gamma$ be an abelian group and $g \\geq h \\geq 2$ be integers. A set $A \\subset \\Gamma$ is a $C_h[g]$-set if given any set $X \\subset \\Gamma$ with $|X| = k$, and any set $\\{ k_1 , \\dots , k_g \\} \\subset \\Gamma$, at least one of the translates $X+ k_i$ is not contained in $A$. For any $g \\geq h \\geq 2$, we prove that if $A \\subset \\{1,2, \\dots ,n \\}$ is a $C_h[g]$-set in $\\mathbb{Z}$, then $|A| \\leq (g-1)^{1/h} n^{1 - 1/h} + O(n^{1/2 - 1/2h})$. We show that for any integer $n \\geq 1$, there is a $C_3 [3]$-set $A \\subset \\{1,2, \\dots , n \\}$ with $|A| \\geq (4^{-2/3} + o(1)) n^{2/3}$. 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