Seminar 2.28.19 Dymek

Title: Proximality of generalized -free systems

Speaker: Aurelia Dymek (Nicolaus Copernicus University)

Abstract: For any subset of integers by -free numbers we call the set of all integers that are not divisible by any element of . A -free system is the orbit closure of the characteristic function of -free numbers under the left shift. The study of -free systems began when Sarnak proposed to deal with dynamical properties of square-free system, i.e., -free system where is the set of all squares of primes. As he postulated this system is proximal. In the joint paper with Kasjan, Kulaga-Przymus and Lemanczyk we showed that a -free system is proximal if and only if contains an infinite pairwise coprime subset. Some multidimensional generalizations of -free systems where studied by Cellarosi, Vinogradov, Baake and Huck.

The topic of my talk is the proximality of generalized -free systems in the case of number fields and lattices. Our main results are the similar characterization of proximality in case of number fields and some lattices. We will give an example that such theorem fails in case of general lattices.