{"paper":{"title":"Haar-$\\mathcal I$ sets: looking at small sets in Polish groups through compact glasses","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GN","authors_text":"Eliza Jab{\\l}o\\'nska, Jaros{\\l}aw Swaczyna, Szymon G\\l\\k{a}b, Taras Banakh","submitted_at":"2018-03-18T18:46:48Z","abstract_excerpt":"Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$\\mathcal I$ set in a Polish group. Here $\\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\\subset X$ of a Polish group $X$ is called Haar-$\\mathcal I$ if there exists a continuous map $f:K\\to X$ such that $f^{-1}(B+x)\\in\\mathcal I$ for all $x\\in X$. Moreover, $B$ is generically Haar-$\\mathcal I$ if the set of witness functions $\\{f\\in C(K,X):\\forall x\\in X\\;\\;f^{-1}(B+x)\\in\\mathcal I\\}$ is comeager in the function sp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06712","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1803.06712/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}