October 2017

Ziemowit Kostana: Non-measurabity of algebraic sum

Tuesday, October 17, 2017 17:15

Room: D1-215

Speaker:
Ziemowit Kostana

Title: Non-measurabity of algebraic sum

Abstract. Consider following problems:

1. If $$A$$ is meagre (null) subset of real line, does there necessarily exist set $$B$$ such that algebraic sum $$A+B$$ doesn't have Baire property (is non-measurable)?

2. If $$A$$ is meagre (null) subset of real line, does there necessarily exist non-meagre (non-null) additive subgroup, disjoint with some translation of $$A$$?

It is not hard to prove that positive answer to 2. implies positive answer to 1, both for measure and category. We answer 2. affirmatively for category, while version for measure turns out to be independent of ZFC. The latter was essentially proved last year by A. Rosłanowski and S. Shelah. Both results holds for Cantor space with coordinatewise addition mod. 2 as well.

Aleksander Cieślak: Ideals of subsets of plane

Tuesday, October 10, 2017 17:15

Room: D1-215

Speaker:
Aleksander Cieślak

Title: Ideals of subsets of plane

Abstract. For given two ideals I and J of subsets of Polish space X we define a Fubini product $$I \times J$$ as all these subsets of plane $$X^2$$ which can be covered by a Borel set B such that I-almost all its vertical sections are J-small. We will investigate how properties of factors influence properties of product.