arXiv:2607.11330v3

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

Francesco Cellarosi

math.DSmath.NT37A3537A4437C8537B4037B0511N2511R0428D15

Paper metadata

Original submission: July 13, 2026
Latest revision: July 24, 2026 · version 3
Claimed MSC: 37A35, 37A44, 37C85, 37B40, 37B05, 11N25, 11R04, 28D15
arXiv categories: math.DS, math.NT
Paper license: CC BY 4.0 ↗
Comments: 14 pages

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.

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Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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01Statements3 reported findingsCorrect

The intrinsic-ergodicity theorem and its applications are supported. Theorem 1.3 identifies a unique maximal-entropy measure $\kappa$ , computes its entropy, and distinguishes it from the Mirsky measure. Corollary 1.4 transfers intrinsic ergodicity to the four listed lattice and number-field systems. Corollary 6.1 additionally gives equidistribution of uniformly random admissible finite patterns toward $\kappa$ .

Theorem 1.3Correct

Intrinsic ergodicity and explicit maximal-entropy measure

Pages 5–7 · Theorem 1.3 and Corollary 1.4 · arXiv:2607.11330v3

For every Erdős family of ideals in the ring of integers of a number field, the $B$ -free subshift has a unique measure of maximal entropy. The measure $\kappa$ is described explicitly: first choose the omitted-residue phase with Haar law, then place independent fair bits on its allowed positions and zeros elsewhere. The conditional entropy is one bit per allowed site, yielding the displayed product density. The density of occupied allowed sites is one half under $\kappa$ and one under the Mirsky measure, proving the asserted mutual singularity.

Corollary 1.4Correct

$k$ -free and $B$ -free lattice/number-field special cases

Page 4 · Corollary 1.4 · arXiv:2607.11330v3

The cited constructions realize each listed $k$ -free lattice-point, $B$ -free lattice-point, or $k$ -free number-field subshift either as an instance of $X_B$ for an Erdős family or as a topologically conjugate system. Intrinsic ergodicity and topological entropy are invariant under those conjugacies, and the pushforward of $\kappa$ gives the unique maximal measure. No extra conclusion is asserted for families outside the Erdős hypotheses.

Corollary 6.1Correct

Equidistribution of uniformly random admissible patterns

Pages 22–23 · Corollary 6.1 · arXiv:2607.11330v3

Uniform measures on admissible patterns over a Følner exhaustion have entropy per site converging to the topological entropy. Every weak limit is invariant and maximal, so Theorem 1.3 forces the unique limit $\kappa$ . Tightness is automatic on the compact subshift. Therefore the entire sequence, not merely a selected subsequence, converges to $\kappa$ as stated.

02Proofs3 reported findingsCorrect

Haar phase, conditional entropy, and exact tiling. Every maximal-entropy measure is first concentrated on the saturated set $Y$ , on which each ideal modulus has exactly one omitted residue. The phase map $\theta$ is Borel and equivariant and sends any candidate maximizer to Haar measure on the profinite rotation. Abramov–Rokhlin decomposes entropy over this zero-entropy factor. At each allowed coordinate the conditional entropy is at most one bit and at forbidden coordinates it is zero. Equality with topological entropy forces equality in all these bounds. Proposition 3.4 upgrades the one-site equalities to conditional independence, identifying the full fiber law with independent fair coins and hence the measure with $\kappa$ .

Relative-entropy rigidityCorrect and complete

Haar phase, conditional entropy, and exact tiling

Pages 12–23 · Sections 3–6 · arXiv:2607.11330v3

Every maximal-entropy measure is first concentrated on the saturated set $Y$ , on which each ideal modulus has exactly one omitted residue. The phase map $\theta$ is Borel and equivariant and sends any candidate maximizer to Haar measure on the profinite rotation. Abramov–Rokhlin decomposes entropy over this zero-entropy factor. At each allowed coordinate the conditional entropy is at most one bit and at forbidden coordinates it is zero. Equality with topological entropy forces equality in all these bounds. Proposition 3.4 upgrades the one-site equalities to conditional independence, identifying the full fiber law with independent fair coins and hence the measure with $\kappa$ .

Sections 3–4Correct and complete

Phase projection and entropy decomposition

Pages 10–17 · Sections 3–4 · arXiv:2607.11330v3

The omitted residue is unique on $Y$ , making $\theta$ well defined coordinatewise and Borel. Equivariance sends invariant measures to invariant measures of the profinite rotation, whose relevant maximal factor is Haar. The skew-product representation $\Phi$ records a phase and coin field, and Abramov–Rokhlin splits entropy into the zero-entropy base plus conditional fiber entropy. Boundary terms vanish along the chosen Følner sets.

Sections 5–6Correct and complete

Rigidity of equality and uniqueness

Pages 17–23 · Sections 5–6 · arXiv:2607.11330v3

The entropy ordering exposes one new allowed coordinate at a time. Equality in the binary entropy bound forces probability one half conditionally on the preceding coordinates; exact tilings show that the ordering samples every allowed site with the correct density. Iteration yields independent fair conditional coordinates over almost every phase. Thus all maximal measures have the same disintegration as $\kappa$ . The final entropy and singularity computations follow directly from this disintegration.

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No non-novelty findings.

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Paper: arXiv:2607.11330v3
Authors listed: Francesco Cellarosi
Audit date: August 18, 2026

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

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Intrinsic ergodicity and explicit maximal-entropy measure

kkk-free and BBB-free lattice/number-field special cases

Equidistribution of uniformly random admissible patterns

Haar phase, conditional entropy, and exact tiling

Phase projection and entropy decomposition

Rigidity of equality and uniqueness

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

$k$ -free and $B$ -free lattice/number-field special cases