{"paper":{"title":"Haar-smallest sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Adam Kwela","submitted_at":"2017-11-22T21:45:48Z","abstract_excerpt":"In this paper we are interested in the following notions of smallness: a subset $A$ of an abelian Polish group $X$ is called Haar-countable/Haar-finite/Haar-$n$ if there are a Borel hull $B\\supseteq A$ and a copy $C$ of $2^\\omega$ such that $(C+x)\\cap B$ is countable/finite/of cardinality at most $n$, for all $x\\in X$.\n  Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar-$\\mathcal{I}$ sets, where $\\mathcal{I}$ is a collection of subsets of $2^\\omega$. It turns out that if $\\mathcal{I}$ is the $\\sigma$-ideal of countable sets, the ideal of fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.09753","kind":"arxiv","version":4},"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"}