{"paper":{"title":"Some properties of $\\mathcal{I}$-Luzin sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Marcin Michalski, Szymon \\.Zeberski","submitted_at":"2015-01-20T18:06:00Z","abstract_excerpt":"In this paper we consider a notion of $\\mathcal{I}$-Luzin set which generalizes the classical notion of Luzin set and Sierpi{\\'n}ski set on Euclidean spaces. We show that there is a translation invariant $\\sigma$-ideal $\\mathcal{I}$ with Borel base for which $\\mathcal{I}$-Luzin set can be $\\mathcal{I}$-measurable. If we additionally assume that $\\mathcal{I}$ has Smital property (or its weaker version) then $\\mathcal{I}$-Luzin sets are $\\mathcal{I}$-nonmeasurable. We give some constructions of $\\mathcal{I}$-Luzin sets involving additive structure of $\\mathbb{R}^n$. Moreover, we show that if $L$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04900","kind":"arxiv","version":2},"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"}