Ergodic properties of m-free integers in number fields

-
Francesco Cellarosi , University of Illinois, Urbana-Champaign
Fine Hall 601

PLEASE NOTE SPECIAL DATE.  For an arbitrary number field $K/Q$ of degree $d$, we study the n-point correlations for $m$-free integers in the ring $O_K$ and define an associated natural $O_K$-action. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $Z^d$ action on a compact abelian group. As a corollary, we obtain that this natural action is not weakly mixing and has zero measure-theoretical entropy. The case $K=Q$, was studied by Ya.G. Sinai and myself, and our theorem provides a different proof to a result by P. Sarnak. This is a joint work with I. Vinogradov.