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Let $\\mathsf{s}'_t(I_k)$ be the smallest positive integer $\\ell$ such that every zero-sum sequence $S$ over $I_k$ of length $|S|\\ge \\ell$ contains a zero-sum subsequence of length $t$. If no such $\\ell$ exists, then let $\\mathsf{s}'_t(I_k)=\\infty$.\n  In this paper, we prove that $\\mathsf{s}'_t(I_k)$ is finite if and only if every integer in $[1,D(I_k)]$ divides $t$, where $D(I_k)=\\max\\{2,2k-1\\}$ is the Davenport constant of $I_k$. Moreover, we prove that if $\\mathsf{s}'_t(I_k)$ is finite, then $t+k(k-1"},"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":"1603.03978","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-13T01:02:31Z","cross_cats_sorted":[],"title_canon_sha256":"f6fb5811c750c6af64e91c2b3d3c6ba7cc21ff45ac29bc1bfe24eba6b34a331d","abstract_canon_sha256":"a193e44f522336d3eb5fd9723a7cabbbfb1079518761539ff74ff13940872fcf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:06.413116Z","signature_b64":"gwPiMdMvFb9vLDZ/uu06ZQQvOGxhAZxHgl8BApwqipi4afvYszwCTy25tNvwevSVrqZHm+iffsJCwRC03ep6AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d06414c297d46703eb3ac39f53cf0c7a54ee8ccf7f3b660e5b0304d3672c926","last_reissued_at":"2026-05-18T00:56:06.412673Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:06.412673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Avoiding zero-sum subsequences of prescribed length over the integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C. Augspurger, K. Shoukry, K. Voss, M. Minter, P. Sissokho","submitted_at":"2016-03-13T01:02:31Z","abstract_excerpt":"Let $t$ and $k$ be a positive integers, and let $I_k=\\{i\\in \\mathbb{Z}:\\; -k\\leq i\\leq k\\}$. Let $\\mathsf{s}'_t(I_k)$ be the smallest positive integer $\\ell$ such that every zero-sum sequence $S$ over $I_k$ of length $|S|\\ge \\ell$ contains a zero-sum subsequence of length $t$. If no such $\\ell$ exists, then let $\\mathsf{s}'_t(I_k)=\\infty$.\n  In this paper, we prove that $\\mathsf{s}'_t(I_k)$ is finite if and only if every integer in $[1,D(I_k)]$ divides $t$, where $D(I_k)=\\max\\{2,2k-1\\}$ is the Davenport constant of $I_k$. Moreover, we prove that if $\\mathsf{s}'_t(I_k)$ is finite, then $t+k(k-1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03978","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.03978","created_at":"2026-05-18T00:56:06.412733+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.03978v3","created_at":"2026-05-18T00:56:06.412733+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.03978","created_at":"2026-05-18T00:56:06.412733+00:00"},{"alias_kind":"pith_short_12","alias_value":"PUDECTBJPVDH","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"PUDECTBJPVDHAPVT","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"PUDECTBJ","created_at":"2026-05-18T12:30:39.010887+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.08237","citing_title":"On small balanceable, strongly-balanceable and omnitonal graphs","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6","json":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6.json","graph_json":"https://pith.science/api/pith-number/PUDECTBJPVDHAPVTVQ47KPHQY6/graph.json","events_json":"https://pith.science/api/pith-number/PUDECTBJPVDHAPVTVQ47KPHQY6/events.json","paper":"https://pith.science/paper/PUDECTBJ"},"agent_actions":{"view_html":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6","download_json":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6.json","view_paper":"https://pith.science/paper/PUDECTBJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.03978&json=true","fetch_graph":"https://pith.science/api/pith-number/PUDECTBJPVDHAPVTVQ47KPHQY6/graph.json","fetch_events":"https://pith.science/api/pith-number/PUDECTBJPVDHAPVTVQ47KPHQY6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6/action/storage_attestation","attest_author":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6/action/author_attestation","sign_citation":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6/action/citation_signature","submit_replication":"https://pith.science/pith/PUDECTBJPVDHAPVTVQ47KPHQY6/action/replication_record"}},"created_at":"2026-05-18T00:56:06.412733+00:00","updated_at":"2026-05-18T00:56:06.412733+00:00"}