## EVENT DETAILS AND ABSTRACT

**Dynamical Systems Seminar**
**Title:** Ergodic properties of k-free integers in number fields

**Speaker:** Francesco Cellarosi

**Speaker Info:** University of Illinois

**Brief Description:**

**Special Note**:

**Abstract:**

The goal of this talk is to study the "randomness" of the set of k-free integers in an arbitrary number field. This is done by encoding all the statistical properties of this set into one dynamical systems (a group action) and by studying the ergodic properties of this system. It turns out that $k$-free integers have the least possible amount of randomness
In fact, if $K/\mathbb{Q}$ is of degree $d$, then the corresponding dynamical system has pure point spectrum, and is isomorphic to a $Z^d$ action on a compact abelian group. This implies that the system is not weakly mixing and has zero measure-theoretical entropy.
The case $K=/\mathbb{Q}$ and $k=2$, was studied previously by Ya.G. Sinai and myself, and our theorem provides a different proof to a result by P. Sarnak.
This is joint work with I. Vinogradov (Bristol).

**Date:** Tuesday, October 22, 2013

**Time:** 4:00pm

**Where:** Lunt 105

**Contact Person:** Prof. Laura DeMarco

**Contact email:** demarco@northwestern.edu

**Contact Phone:**

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