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 Erdős 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 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.

Disclosure

“Acknowledgements The author gratefully acknowledges the support of the Natural Sciences and Engineering Re- search Council of Canada through the NSERC Discovery Grants RGPIN-2022-04330. During the preparation of this work the author used Claude (Anthropic, Opus 4.8 and Fable 5 models, accessed via claude.ai) to draft and refine exposition and LaTeX and to assist with liter- ature and reference searches. All mathematical results, proofs, and conclusions are the author’s own. The a”

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2026-08-23T18:51:24.190130Z

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