arXiv:2607.11178v1

Zero-one laws for uniform approximation via Gaussian and Eisenstein integers

René Pfitscher, Anurag Rao, Shucheng Yu, Han Zhang

math.DSmath.NT37A1711J1311H06

Paper metadata

Original submission: July 13, 2026
Claimed MSC: 37A17, 11J13, 11H06
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 44 pages, comments welcome

Abstract

We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing to a shrinking target problem on certain homogeneous spaces of $\mathrm{SL}_2(\mathbb{C})$ . The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.

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Audited against arXiv v1

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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Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The Dirichlet theorem for imaginary quadratic fields, the Gaussian and Eisenstein zero-one laws, and the critical-locus neighborhood estimates are correct as stated.

Theorem 1.1Correct

The imaginary-quadratic Dirichlet constant is correctly characterized and bounded

Pages 2 and 13–15 · Theorem 1.1 and Proposition 2.4 · arXiv:2607.11178v1

Dani's correspondence identifies uniform approximation with avoidance of the critical-radius cusp neighborhood. Ergodicity rules out improvement below the critical radius for almost every parameter, while compactness gives the universal upper bound. The covolume calculation and the two explicit lattice constructions give the displayed discriminant bounds, and Minkowski's critical-locus descriptions give the Gaussian and Eisenstein constants used later.

Theorem 1.2Correct

The two zero-one criteria have the stated series and orientations

Pages 3–4 and 32–43 · Theorem 1.2 and Sections 5.4–5.5 · arXiv:2607.11178v1

With $F_\psi(t)=(c_K-t\psi(t))/c_K$ , Proposition 5.1 converts failure of uniform approximation to visits to the shrinking critical-locus targets. The target measures are comparable to $F_\psi(k)^4$ in the Gaussian case and to $F_\psi(k)^2\log(1/F_\psi(k))$ in the Eisenstein case. The convergence argument and the correlation estimate in the divergent argument therefore yield exactly the two displayed zero-one laws.

Theorem 1.3Correct

The critical-locus neighborhood asymptotics are correct

Page 5 and pages 16–29 · Theorem 1.3, Propositions 3.3 and 3.6 · arXiv:2607.11178v1

In the Gaussian case the local determinant constraint has four independent small real directions, giving order $r^4$ . In the Eisenstein case the singular stratum contributes the logarithmic integral and gives order $r^2\log(1/r)$ . The coordinate charts, Jacobians, exceptional sets, and matching upper and lower inclusions establish both two-sided estimates uniformly for sufficiently small $r$ .

02Proofs3 reported findingsCorrect

The proofs of the central statements and their material dynamical and geometric inputs are correct and complete. The short-range contribution is handled by genuine disjointness estimates rather than by an unsupported independence assumption.

Section 3Correct and complete

Critical-locus charts and target-measure estimates

Pages 16–29 · Propositions 3.3, 3.6, and 3.10 · arXiv:2607.11178v1

The paper proves both inclusions between the geometric target and explicit coordinate inequalities, controls the noncompact remainder by injectivity-radius estimates, and integrates the resulting regions. Proposition 3.10 then supplies the uniform estimates needed for smoothing and for the later short-range intersections.

Section 4Correct and complete

Single and double effective equidistribution

Pages 29–32 · Theorems 4.1 and 4.2 · arXiv:2607.11178v1

The cited expanding-horosphere estimate has error terms controlled by a positive power of the injectivity radius; the manuscript's simplified exponent follows after absorbing the polynomial time factor and using that the spectral exponent is at most one. The self-contained double-correlation argument removes the thin part, applies single equidistribution on the thick part, and balances the cutoff so that the displayed exponential error follows.

Section 5Correct and complete

Dani correspondence and the two Borel–Cantelli directions

Pages 32–43 · Proposition 5.1, Lemmas 5.2–5.6, and Sections 5.4–5.5 · arXiv:2607.11178v1

The discrete target formulation preserves the monotonicity assumptions and compares the integral and series criteria correctly. The convergence half uses the first Borel–Cantelli lemma. For divergence, the reduction lemma preserves divergence, long-range correlations are controlled by effective mixing, and the remaining logarithmic window is bounded by the paper's pairwise-disjoint target construction; the resulting second-moment bound is sufficient for the divergent Borel–Cantelli conclusion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.11178v1
Authors listed: René Pfitscher, Anurag Rao, Shucheng Yu, Han Zhang
Audit date: August 15, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. 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Zero-one laws for uniform approximation via Gaussian and Eisenstein integers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The imaginary-quadratic Dirichlet constant is correctly characterized and bounded

The two zero-one criteria have the stated series and orientations

The critical-locus neighborhood asymptotics are correct

Critical-locus charts and target-measure estimates

Single and double effective equidistribution

Dani correspondence and the two Borel–Cantelli directions

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