Ergodic properties of $k$-free integers in number fields
Francesco Cellarosi - Department of Mathematics, Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, United States (email) Abstract:
Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers
$\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a
natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences,
equipped with an $\mathscr O_K$-invariant probability measure associated to
$\mathscr F_k$. We prove that this action is ergodic, has pure point
spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian
group. In particular, it is not weakly mixing and has zero
measure-theoretical entropy. This work generalizes the work of
Cellarosi and Sinai [
Keywords: Square-free and $k$-free integers in number fields,
correlation functions, group actions with pure point spectrum,
ergodicity, isomorphism of group actions.
Received: March 2013; Revised: September 2013; Available Online: December 2013. |