{"paper":{"title":"On products of k atoms II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alfred Geroldinger, David J. Grynkiewicz, Pingzhi Yuan","submitted_at":"2015-03-20T17:11:26Z","abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor (for example, rings of integers in algebraic number fields or holomorphy rings in algebraic function fields). For $k \\in \\mathbb N$, let $\\mathcal U_k (H)$ denote the set of all $m \\in \\mathbb N$ with the following property: There exist atoms $u_1, ..., u_k, v_1, ..., v_m \\in H$ such that $u_1 \\cdot ... \\cdot u_k = v_1 \\cdot ...\\cdot v_m$. Furthermore, let $\\lambda_k (H) = \\min \\mathcal U_k (H)$ and $\\rho_k (H) = \\sup \\mathcal U_k (H)$. The sets $\\mathcal U_k (H) \\subset \\mathbb N$ are intervals which"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06164","kind":"arxiv","version":1},"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"}