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The set of distances $\\Delta (H)$ of $H$ is the set of all $d \\in \\mathbb{N}$ with the following property: there are irreducible elements $u\\_1, \\ldots, u\\_k, v\\_1 \\ldots, v\\_{k+d}$ such that $u\\_1 \\cdot \\ldots \\cdot u\\_k=v\\_1 \\cdot \\ldots \\cdot v\\_{k+d}$ but $u\\_1 \\cdot \\ldots \\cdot u\\_k$ cannot be written as a product of $\\ell$ irreducible elements for any $\\ell \\in \\mathbb{N}$ with $k\\lt \\ell \\lt k+d$. It is well-known (and easy to show) that, if $\\Delta (H)$ is nonempty, then   $\\min \\Delta (H) = \\gcd \\Delta (H)$. In this paper we show conversely that for every"},"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":"1608.06407","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-08-23T07:40:01Z","cross_cats_sorted":[],"title_canon_sha256":"1dc3fb5254e801870d3cd8ea09ebc62dfdaf4ccede30b8407ee80cbbd7d29a26","abstract_canon_sha256":"156c380bb791e6f9eaec6faa03ec77d3a62257b01bc73b45a66361afa889889a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:34.133707Z","signature_b64":"3OlVUdzk+VTvrW0zeYxCWKAqi+14rEhYpSe0z3CfrqSTltrLJQjBK0sLGzMkjn3LA117F9xxovb0gq222mikAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fdce2e7b8a2672b7e332553983efc2f70a32a08c5829705abc3e7eed5b91d62","last_reissued_at":"2026-05-18T00:52:34.133256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:34.133256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A realization theorem for sets of distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger (IM), Wolfgang Schmid (LAGA)","submitted_at":"2016-08-23T07:40:01Z","abstract_excerpt":"Let $H$ be an atomic monoid. The set of distances $\\Delta (H)$ of $H$ is the set of all $d \\in \\mathbb{N}$ with the following property: there are irreducible elements $u\\_1, \\ldots, u\\_k, v\\_1 \\ldots, v\\_{k+d}$ such that $u\\_1 \\cdot \\ldots \\cdot u\\_k=v\\_1 \\cdot \\ldots \\cdot v\\_{k+d}$ but $u\\_1 \\cdot \\ldots \\cdot u\\_k$ cannot be written as a product of $\\ell$ irreducible elements for any $\\ell \\in \\mathbb{N}$ with $k\\lt \\ell \\lt k+d$. It is well-known (and easy to show) that, if $\\Delta (H)$ is nonempty, then   $\\min \\Delta (H) = \\gcd \\Delta (H)$. 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