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arXiv:1609.06780v3

A zero-one Law for improvements to Dirichlet's Theorem

Dmitry Kleinbock, Nick Wadleigh

math.NTmath.DS11J0411J7037A17
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Paper metadata

Original submission
September 21, 2016
Latest revision
February 17, 2017 · version 3
Claimed MSC
11J04, 11J70, 37A17
arXiv categories
math.NT, math.DS
Comments
12 pages; minor corrections made, Corollary 3.7 added to the latest version

Abstract

We give an integrability condition on a function ψψ guaranteeing that for almost all (or almost no) x∈Rx\in\mathbb{R}, the system ∣qx−p∣≤ψ(t)|qx-p|\leq ψ(t), ∣q∣<t|q|<t is solvable in p∈Zp\in \mathbb{Z}, q∈Z∖{0}q\in \mathbb{Z}\smallsetminus \{0\} for sufficiently large tt. Along the way, we characterize such xx in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.

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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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Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

The non-improvability result and the zero--one Lebesgue criterion for improvements to one-dimensional Dirichlet approximation are correct.

Theorem 1.7Correct

Every strict eventual improvement has a counterexample

Pages 2 and 5 · Theorem 1.7 and continued-fraction construction · arXiv:1609.06780v3

The continued-fraction criterion rewrites ψ\psi-Dirichlet improvability as an eventual upper bound on an+1qnqn+1a_{n+1}q_nq_{n+1}. Because tψ(t)<1t\psi(t)<1 eventually, the reciprocal gap function tends above one along the required scales. Choosing partial quotients recursively to exceed that threshold infinitely often constructs an irrational outside D(ψ)D(\psi), while preserving a valid continued fraction at every stage.

Theorem 1.8Correct

The convergence--divergence series gives the stated zero--one law

Pages 2 and 9 · Theorem 1.8 and proof · arXiv:1609.06780v3

With Ψ(t)=(1−tψ(t))−1\Psi(t)=(1-t\psi(t))^{-1}, the continued-fraction criterion reduces failure of improvability to anan+1>Ψ(qn)a_na_{n+1}>\Psi(q_n) infinitely often. The Gauss-map Borel--Cantelli theorem has threshold sum comparable to ∑nlog⁡Ψ(n)/Ψ(n)\sum_n\log\Psi(n)/\Psi(n), which is exactly ∑n−log⁡(1−nψ(n))(1−nψ(n))n.\sum_n\frac{-\log(1-n\psi(n))(1-n\psi(n))}{n}. Exponential bounds for qnq_n and condensation transfer between index and denominator without changing convergence.

02Proofs2 reported findingsCorrect

The continued-fraction equivalences, Gauss-measure estimates, mixing Borel--Cantelli argument, and denominator-index transfer are correct and complete.

Section 2Correct and complete

The continued-fraction reformulation is exact

Pages 3--5 · Lemmas 2.1--2.2 · arXiv:1609.06780v3

For qn≤t<qn+1q_n\leq t<q_{n+1}, the best possible value of ⟨qα⟩\langle q\alpha\rangle with q<tq<t is attained at the appropriate convergent. Substituting the standard inequalities for ∣qnα−pn∣|q_n\alpha-p_n| yields the stated threshold involving an+1a_{n+1} and the reciprocal gap. Monotonicity of tψ(t)t\psi(t) is used in the correct direction at both endpoints.

Sections 3--4Correct and complete

The dependent Borel--Cantelli estimate and condensation are valid

Pages 5--12 · Theorem 3.6, Corollary 3.7, and proof of Theorem 1.8 · arXiv:1609.06780v3

The Gauss map has the required exponential mixing for cylinder-separated events. Direct integration gives measure comparable to log⁡Ψ(n)/Ψ(n)\log\Psi(n)/\Psi(n) for the product of adjacent partial quotients. The almost-sure exponential upper and lower bounds for qnq_n, together with monotonicity, compare the threshold sums by Cauchy condensation in both directions.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1609.06780v3
Authors listed
Dmitry Kleinbock, Nick Wadleigh
Audit date
August 19, 2026
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