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

Technical Summary: Intrinsic Ergodicity for B-free Integers in Number Fields

Problem Statement
The paper addresses the problem of intrinsic ergodicity for $B$-free subshifts in the context of number fields. While the structural properties of $B$-free systems (genericity, topological entropy, admissibility, and proximality) have been established for various settings—including the classical squarefree integers ($K=\mathbb{Q}$), $k$-free lattice points, and $B$-free lattice points—the uniqueness of the measure of maximal entropy (intrinsic ergodicity) remained an open question for multidimensional and number-field settings.

Specifically, let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $\mathcal{B} = \{b_\ell\}_{\ell \ge 1}$ be an Erdős family of ideals (infinite, pairwise coprime, with $\sum 1/N(b_\ell) < \infty$). The associated $B$-free subshift $(X_\mathcal{B}, (S_a)_{a \in \mathcal{O}_K})$ is a hereditary system with positive topological entropy. It is known that the system carries a "Mirsky measure" $\nu_\eta$ (the push-forward of the Haar measure on a compact group of rotations), which has zero Kolmogorov-Sinai entropy. The central question is whether there exists a unique invariant measure that maximizes entropy, and if so, to identify it explicitly.

Methodology
The proof proceeds through a series of reductions followed by two independent proofs of a core rigidity statement.

1. Reduction to a Saturated Set and Factor Map:
   Drawing on the work of Araújo, Dymek, and Kułaga-Przymus [1], the authors establish that any measure of maximal entropy is concentrated on a saturated set $Y \subset X_\mathcal{B}$. This set consists of configurations whose support omits exactly one residue class modulo each ideal $b_\ell$.
   A Borel, equivariant "phase map" $\theta: Y \to G$ is defined, where $G = \prod_{\ell \ge 1} \mathcal{O}_K/b_\ell$ is a compact abelian group. The system $(G, T)$ under the rotation action is uniquely ergodic with zero entropy. By the Abramov–Rokhlin addition formula for amenable group actions, the entropy of any invariant measure $\nu$ on $Y$ projecting to the Haar measure $P$ on $G$ is equal to its relative entropy over the rotation factor: $h_\nu(S) = h_\nu(S \mid \mathcal{F}_\theta)$.

2. The Rigidity Statement:
   The problem reduces to proving that among all measures projecting to $P$, the relative entropy is bounded by the density of allowed positions $D = \prod (1 - 1/N(b_\ell))$, with equality holding if and only if the measure is a specific measure $\kappa$.

3. Two Independent Proofs of Rigidity:
   The authors provide two distinct proofs for this rigidity statement:

   • Relative Entropy Argument (Section 4): This proof utilizes a single-site conditional entropy approach. By fixing a lexicographic order on $\mathcal{O}_K \cong \mathbb{Z}^d$, the relative entropy is expressed as the conditional entropy of a single site given its past and the phase. The authors show that maximizing this entropy forces the conditional probability of a site being "allowed" (1) to be exactly $1/2$ (a fair coin) whenever the site is not forbidden by the phase. This uniquely determines the measure as a product of independent fair coins on the allowed set.
   • Induce-and-Split via Exact Tiling (Section 5): This proof adapts Peckner's induce-and-split scheme [16] to the multidimensional setting. Since $\mathcal{O}_K$ lacks a first-return map (unlike $\mathbb{Z}$), the authors replace induction with an exact tiling of $\mathcal{O}_K$ by cubes. They define a "super-symbol" process on the sublattice of tile origins. The proof demonstrates that at maximal entropy, the relative entropy of the system equals the entropy of the super-symbols, which in turn forces the super-symbols to be conditionally independent and uniformly distributed on the allowed configurations within each tile. This structure implies that the measure is a relatively independent extension of the base rotation with Bernoulli fibers.

Key Contributions and Results

• Theorem 1.3 (Main Result): The $B$-free subshift $(X_\mathcal{B}, S)$ is intrinsically ergodic. It possesses a unique measure of maximal entropy, denoted $\kappa$.

  • Explicit Identification: $\kappa$ is constructed as the push-forward of the product measure $P \otimes \beta$ (where $P$ is Haar on $G$ and $\beta$ is the uniform Bernoulli measure on $\{0,1\}^{\mathcal{O}_K}$) via a coding map $\Phi(g, \omega) = \phi(g) \cdot \omega$.
  • Entropy Value: The topological entropy is $h_{top}(X_\mathcal{B}) = \prod_{\ell \ge 1} (1 - 1/N(b_\ell))$. The measure $\kappa$ achieves this entropy.
  • Structure: Conditioned on a Haar-random phase $g \in G$, $\kappa$ places independent fair coins on the allowed positions $A_g$ (those not congruent to $-g_\ell \pmod{b_\ell}$) and zeros elsewhere.
  • Singularity: $\kappa$ is mutually singular with the Mirsky measure $\nu_\eta$. While $\nu_\eta$ corresponds to the "all-ones" configuration on allowed sites (zero entropy), $\kappa$ corresponds to a random configuration (positive entropy).

• Corollary 1.4 (Generalizations): As a consequence of Theorem 1.3 and the fact that intrinsic ergodicity is a topological conjugacy invariant, the paper establishes intrinsic ergodicity for:

  • $k$-free lattice-point subshifts (Pleasants–Huck, Huck–Baake).
  • $B$-free lattice-point subshifts (Baake–Huck).
  • $k$-free number-field subshifts (Cellarosi–Vinogradov).

• Equidistribution (Corollary 6.1): The paper proves that uniformly random admissible patterns on large Følner boxes equidistribute according to the measure $\kappa$.

Significance and Claims
The paper claims to provide the first proof of intrinsic ergodicity for $B$-free systems beyond dimension one. It resolves the open problem for $B$-free integers in number fields and related lattice settings.

The authors highlight the "genuine rigidity" of the result: despite the system being hereditary and supporting a large simplex of invariant measures, the arithmetic structure forces a unique "most chaotic" measure. This measure is structurally distinct from the arithmetically natural Mirsky measure (which is a rotation), yet it is intimately related to it as a relatively independent extension. The work demonstrates that the "order" of the arithmetic rotation and the "chaos" of the maximal entropy measure can coexist, with the latter being the unique maximizer of entropy. The proof techniques, particularly the adaptation of the induce-and-split scheme to multidimensional settings via exact tilings, are presented as significant methodological contributions.