2015/09/30 by Тарас Банах, Taras Banakh, Banakh, Taras +4
Mathematics · #22A15 #54H11 #54H99 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Rings, Modules, and Algebras #math.GN #math.GR #msc:22A15 #msc:54H11 #msc:54H99
paper · pdf · doi:10.48550/arxiv.1509.09073
18 pages
arxiv created 2015/09/30 · openalex publication_date 2015/09/30 · arxiv updated 2015/10/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We prove that any meager quasi-analytic subgroup of a topological group G belongs to every σ-ideal \mathcal I on G possessing the closed ± n-Steinhaus property for some n∈\mathbb N. An ideal \mathcal I on a topological group G is defined to have the closed ± n-Steinhaus property if for any closed subsets A1,…,An∉\mathcal I of G the product (A1∪ A1-1)⋯ (An∪ An-1) is not nowhere dense in G. Since the σ-ideal \mathcal E generated by closed Haar null sets in a locally compact group G has the closed ± 2-Steinhaus property, we conclude that each meager quasi-analytic subgroup H⊂ G belongs to the ideal \mathcal E. For analytic subgroups of the real line this result was proved by Laczkovich in 1998. We shall discuss possible generalizations of the Laczkovich Theorem to non-locally compact groups and construct an example of a meager Borel subgroup in \mathbb Zω which cannot be covered by countably many closed Haar-null (or even closed Haar-meager) sets. On the other hand, assuming that cof(\mathcal M)=cov(\mathcal M)=cov(\mathcal N) we construct a subgroup H⊂ 2ω which is meager and Haar null but does not belong to the σ-ideal \mathcal E. The construction uses a new cardinal characteristic voc^*(\mathcal I,\mathcal J) which seems to be interesting by its own.