Jarosław Swaczyna

Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017 17:15

Room: D1-215

Speaker:
Jarosław Swaczyna

Title: Haar-small sets

Abstract. In locally compact Polish groups there is a very natural \(\sigma\)-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.

Jarosław Swaczyna: Generalized densities of subsets of natural numbers and associated ideals

Tuesday, May 19, 2015 17:15

Room: D1-215

Speaker:
Jarosław Swaczyna

Title: Generalized densities of subsets of natural numbers and associated ideals

Abstract. Let \(g: \omega \rightarrow [0, \infty)\). We say that \(A \subset \omega\) has \(g\)-density zero, if \(\lim_{n \rightarrow \infty} \frac{A \cap n}{g(n)} = 0\). It is an easy observation that family of \(g\)-density zero sets is an ideal.

I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions \(g\).

I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.