Continued Fractions, Interval Exchanges and Applications to Geometry

Ergodic properties of m-free integers in number fields

speaker: Francesco Cellarosi (Queen's University)

abstract: It is known that not every number field is a Euclidean domain, but unique factorization holds for ideals. Using this fact we introduce the notion of $m$-free integers. For an arbitrary number field $KQ$ 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.

timetable:
Tue 11 Jun, 14:00 - 14:50, Aula Dini
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