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In this paper, among other things, we give estimates for $s_A(C_3^r)$, and prove that $s_A(C_{3}^{3})=9$, $s_A(C_{3}^{4})=21$ and $41\\leq s_A(C_{3}^{5})\\leq45$."},"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":"1201.0276","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-31T15:56:41Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2c24ea6aa7864ac97676e6111600c2bbd92564df752d2b0a04a588a67c25a444","abstract_canon_sha256":"e6c73a604a37ce9f0d7651951923929036bdcec12470925e46c2d4b89e8e32af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:25.568736Z","signature_b64":"B4UhljFYcEl10wZ/MfCmleBvJgd+5E99AA6qIt63LEFGJsVm+GCYCDF8JWUTbevr6sM5Pds3OYOH1EimhZS/Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a9483947ee260100c8c7752c304907ad40f91142731b5b343ead57142d23f7a","last_reissued_at":"2026-05-18T04:05:25.568121Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:25.568121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted Zero-Sum Problems Over $C_3^r$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ab\\'ilio Lemos, Diego Marques, Hemar Godinho","submitted_at":"2011-12-31T15:56:41Z","abstract_excerpt":"Let $C_n$ be the cyclic group of order $n$ and set $s_{A}(C_n^r)$ as the smallest integer $\\ell$ such that every sequence $\\mathcal{S}$ in $C_n^r$ of length at least $\\ell$ has an $A$-zero-sum subsequence of length equal to $\\exp(C_n^r)$, for $A=\\{-1,1\\}$. 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