Sunday, September 17, 2017, 17:15

Wrocław University of Technology, 215 D-1

Speaker: Ziemowit Kostana (University of Warsaw)

Title: Non-measurability 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.