2026/07/13 by Francesco Cellarosi
#math.DS #math.NT
Let K be a number field with ring of integers \mathscrOK, and let \mathfrakB be an Erdős family of ideals in \mathscrOK. We prove that the associated \mathfrakB-free subshift (X_\mathfrakB,(Sa)_a∈\mathscrOK) 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 ∏_\mathfrakb∈\mathfrakB\mathscrOK/\mathfrakb. This is the first proof of intrinsic ergodicity for \mathfrakB-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 \mathfrakB-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.