{"paper":{"title":"Factoring groups into dense subsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Igor Protasov, Serhii Slobodianiuk","submitted_at":"2016-02-04T09:34:02Z","abstract_excerpt":"Let $G $ be a group of cardinality $\\kappa>\\aleph_0 $ endowed with a topology $\\tau $ such that $|U|=\\kappa$ for every non-empty $U\\in\\tau$ and $\\tau$ has a base of cardinality $\\kappa$. We prove that $G$ could be factorized $G=AB$ (i.e. each $g\\in G$ has unique representation $g=ab$, $a\\in A$, $b\\in B$) into dense subsets $A,B$, $|A|=|B|=\\kappa$. We do not know if this statement holds for $\\kappa = \\aleph_0$ even if $G$ is a topological group."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01603","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"}