Intrinsic ergodicity for $\mathfrak{B}$-free integers in number fields

Abstract

Let $K$ be a number field with ring of integers $\mathscr{O}_K$, and let $\mathfrak{B}$ be an Erd\H{o}s family of ideals in $\mathscr{O}_K$. We prove that the associated $\mathfrak{B}$-free subshift $(X_{\mathfrak{B}},(S_a)_{a\in\mathscr{O}_K})$ is intrinsically ergodic: it carries a unique measure of maximal entropy, which we identify explicitly as a relatively independent extension of the Haar rotation on $\prod_{\mathfrak{b}\in\mathfrak{B}}\mathscr{O}_K/\mathfrak{b}$. This is the first proof of intrinsic ergodicity for $\mathfrak{B}$-free systems beyond dimension one, and relies on the work of Ara\'ujo--Dymek--Ku\l aga-Przymus. Via their reductions, we also settle the $k$-free and $\mathfrak{B}$-free lattice-point cases and the $k$-free number-field case. We give two independent proofs of the underlying rigidity statement: one through a single-site relative-entropy argument, and one through an exact-tiling realisation of Peckner's induce-and-split scheme.