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Paper metadata

Original submission
October 5, 2026
Claimed MSC
37A25, 11B57, 11H60
arXiv categories
math.DS, math.NT
Comments
19 pages, 1 figure

Abstract

We prove that the Boca--Cobeli--Zaharescu (BCZ) map is strongly mixing with respect to normalized Lebesgue measure on the BCZ triangle.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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.

Current report

Detailed mathematical audit

Generated October 24, 2026
01Statements2 reported findingsNo statement issues found

The zero-discrepancy estimate and the resulting strong mixing of the BCZ map are supported. The proof correctly transfers a primitive-lattice empty-rectangle bound to return-time discrepancies and then uses weak mixing to identify every positive-discrepancy component of a graph-measure limit.

Theorem 1.2No issue found

The zero-discrepancy probability has the stated sharp limiting upper bound

PDF pp. 2 and 9–17 · Propositions 4.1 and 5.1 · arXiv:2610.07347v1

For the rectangle Bn=(cn,1]×(0,ρˉn]B_n=(c_n,1]\times(0,\bar\rho n], primitive Siegel's formula gives EμNn=2u\mathbb E_\mu N_n=2u. The pair formula and determinant-density calculation give Var⁡μ(Nn)≤2u+en\operatorname{Var}_\mu(N_n)\leq2u+e_n with en→0e_n\to0. Lemmas 5.2–5.4 correctly compare DnD_n first with the section count and then, uniformly over all subsets of N0\mathbb N_0, with the Haar-distributed count. The second-moment inequality P(N=0)≤Var⁡(N)/((EN)2+Var⁡(N))\mathbb P(N=0)\leq\operatorname{Var}(N)/((\mathbb EN)^2+\operatorname{Var}(N)) therefore yields lim sup⁡m(Dn=0)≤1/(1+2u)\limsup m(D_n=0)\leq1/(1+2u).

Theorem 1.1No issue found

Every subsequential graph-measure limit is the product measure

PDF pp. 1 and 6–9 · Lemmas 3.1–3.2 · arXiv:2610.07347v1

The discrepancy decomposition has summable tails because EDn→2u\mathbb E D_n\to2u. Each limiting component is invariant under T×TT\times T and has both marginals equal to its mass times mm. Summing its intermediate vertical shifts produces an Id⁡×T\operatorname{Id}\times T-invariant joining, hence a product measure by ergodicity. Weak mixing then forces every component with positive discrepancy to be a scalar multiple of m⊗mm\otimes m. Theorem 1.2 leaves at most 1/(1+2u)1/(1+2u) mass in the zero component; letting u→∞u\to\infty forces the fixed graph limit to equal m⊗mm\otimes m.

02Proofs3 reported findingsNo proof issues found

The graph-measure, primitive-lattice moment, and section-to-suspension arguments are complete. The critical first-moment and pair-formula inputs match the required normalizations, and all limiting steps are uniform in the discrepancy events where uniformity is needed.

Lemma 3.1 and Lemma 3.2No issue found in the reviewed proof

The component invariance, marginal, and joining calculations are valid

PDF pp. 6–9 · equations (11)–(16) · arXiv:2610.07347v1

The exceptional event on which a discrepancy changes under one application of TT has measure at most twice mβj(rβj>1)m_{\beta_j}(r_{\beta_j}>1), which tends to zero by Kac's formula. Domination of both marginals by βj−2m\beta_j^{-2}m justifies passing the discontinuous maps to weak limits. The two decompositions of the graph limit telescope to an Id⁡×T\operatorname{Id}\times T-invariant measure, and domination plus ergodicity of T×TT\times T identifies each positive component without assuming the desired mixing conclusion.

Proposition 4.1No issue found in the reviewed proof

The thin-rectangle mean and variance calculation has the stated constants

PDF pp. 9–13 · equations (17)–(27) · arXiv:2610.07347v1

The primitive pair formula reduces the second factorial moment to 2ζ(2)−1∑k≥1φ(k)I(k)/k2\zeta(2)^{-1}\sum_{k\geq1}\varphi(k)I(k)/k. The determinant pushforward density II is even, decreasing on the positive half-line, supported in [−h,h][-h,h], and has integral V2/2V^2/2 there. Abel summation with ∣∑k≤tφ(k)/k−t/ζ(2)∣≤3+log⁡h\left|\sum_{k\leq t}\varphi(k)/k-t/\zeta(2)\right|\leq3+\log h gives the printed error, while I(0)=2h(1−c+clog⁡c)≤(1−c)2h/cI(0)=2h(1-c+c\log c)\leq(1-c)^2h/c supplies the final bound.

Lemmas 5.2–5.4; Proposition 5.1No issue found in the reviewed proof

The return count and Haar rectangle count are uniformly asymptotically equivalent

PDF pp. 13–17 · arXiv:2610.07347v1

The ergodic theorem confines the relevant return height to a strip whose area tends to zero after the auxiliary tolerance does. The cross-sectional first-moment formula bounds the probability of an extra primitive vector in that strip uniformly in the scaling. Almost invariance of every event {Dn∈J}\{D_n\in J\} removes the suspension roof weight by the mean ergodic theorem, and the top and bottom boundary strips control the horocycle displacement. Taking first n→∞n\to\infty, then the roof truncation to infinity, proves the required uniform total-variation comparison.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.07347v1
Authors listed
Albert Artiles, Andrew Best, Matteo Bordignon
Audit date
October 24, 2026
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