{"paper":{"title":"Rothberger gaps in fragmented ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Diego A. Mej\\'ia, J\\\"org Brendle","submitted_at":"2014-08-31T14:18:46Z","abstract_excerpt":"The~\\emph{Rothberger number} $\\mathfrak{b} (\\mathcal{I})$ of a definable ideal $\\mathcal{I}$ on $\\omega$ is the least cardinal $\\kappa$ such that there exists a Rothberger gap of type $(\\omega,\\kappa)$ in the quotient algebra $\\mathcal{P} (\\omega) / \\mathcal{I}$. We investigate $\\mathfrak{b} (\\mathcal{I})$ for a subclass of the $F_\\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0222","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"}