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

Abstract

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$, and let $\mathfrak{B}$ be an Erdős family of ideals in ${\mathcal{O}}_K$. We prove that the associated $\mathfrak{B}$-free subshift $(X_{\mathfrak{B}},(S_a{)}_{a\in {\mathcal{O}}_K})$ 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 ${\prod }_{\mathfrak{b}\in \mathfrak{B}}{\mathcal{O}}_K/\mathfrak{b}$. This is the first proof of intrinsic ergodicity for $\mathfrak{B}$-free systems beyond dimension one, and relies on the work of Araújo--Dymek--Kułaga-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.

14 pages

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