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If $G$ is finite, then there is a constant $M \\in \\mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \\in \\Delta^* (H)$, where $\\Delta^* (H)$ denotes the set of minimal distances of $H$. We show that $\\max \\Delta^* (H) \\le \\max \\{\\exp (G)-2, \\mathsf r (G)-1\\}$ 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":"1404.2873","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AC","submitted_at":"2014-04-10T16:34:06Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"c875f445aca3c162be353074526934dff5457bd650ee804877311584023aeb50","abstract_canon_sha256":"e76101c9b21b5a6d62259095b7109dd0282d5b9418d8c3cb24f70a38e41002e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:22.425115Z","signature_b64":"vkOTsyQMcwUE3bZIVIVsnykeamHtbxUmZ8LbXc7+gLTwK2GboB+wNwoETdlsLKqx+GdeBhwVs5eWgOD+619MCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b77733ee492c4ce49a365f276001af685782fba03ed874be61af30151dfe46ab","last_reissued_at":"2026-05-17T23:41:22.424492Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:22.424492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The set of minimal distances in Krull monoids","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger, Qinghai Zhong","submitted_at":"2014-04-10T16:34:06Z","abstract_excerpt":"Let $H$ be a Krull monoid with finite class group $G$. 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