Eliza Jabłońska
Eliza Jabłońska: From the Steinhaus property to the Laczkovich one
10/06/22 08:16
Tuesday, June 14, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Eliza Jabłońska (AGH University of Science and Technology)
Title: From the Steinhaus property to the Laczkovich one
Abstract: Let \(X\) be a locally compact Abelian Polish group, \(\mathcal{B}(X)\) be the family of all Borel subsets of \(X\) and \(\mathcal{F}\subset 2^{X}\). We consider the following Steinhaus' type properties:
It is known that the family \(\mathcal{M}\) of all meager sets as well as the family \(\mathcal{N}\) of all sets of Haar measure zero satisfy each of these conditions. We prove that the family \(\mathcal{M}\cap\mathcal{N}\) satisfies \((S^-)\), \((D^+)\), \((D^-)\) although it does not satisfy \((S^+)\). We also show that the \(\sigma\)-ideal \(\sigma\overline{\mathcal{N}}\subset \mathcal{M}\cap\mathcal{N}\) generated by closed sets of Haar measure zero satisfies only \((D^+)\) and \((D^-)\) which leads us to the Laczkovich property. This is joint work with T. Banakh, I. Banakh, Sz. Głąb and J. Swaczyna.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Eliza Jabłońska (AGH University of Science and Technology)
Title: From the Steinhaus property to the Laczkovich one
Abstract: Let \(X\) be a locally compact Abelian Polish group, \(\mathcal{B}(X)\) be the family of all Borel subsets of \(X\) and \(\mathcal{F}\subset 2^{X}\). We consider the following Steinhaus' type properties:
- \((S^+)\): \((A+B)\neq\emptyset\) for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((S^-)\): \(0\in (A-A)\) for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((D^+)\): \(A+B\) is non-meager for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((D^-)\): \(A-A\) is non-meager for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\).
It is known that the family \(\mathcal{M}\) of all meager sets as well as the family \(\mathcal{N}\) of all sets of Haar measure zero satisfy each of these conditions. We prove that the family \(\mathcal{M}\cap\mathcal{N}\) satisfies \((S^-)\), \((D^+)\), \((D^-)\) although it does not satisfy \((S^+)\). We also show that the \(\sigma\)-ideal \(\sigma\overline{\mathcal{N}}\subset \mathcal{M}\cap\mathcal{N}\) generated by closed sets of Haar measure zero satisfies only \((D^+)\) and \((D^-)\) which leads us to the Laczkovich property. This is joint work with T. Banakh, I. Banakh, Sz. Głąb and J. Swaczyna.