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arXiv:1704.03089v1

Hausdorff measure of sets of Dirichlet non-improvable numbers

Mumtaz Hussain, Dmitry Kleinbock, Nick Wadleigh, Bao-Wei Wang

math.NT
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Paper metadata

Original submission
April 11, 2017
Claimed MSC
arXiv categories
math.NT
Comments
15 pages
Journal reference
Mathematika 64 (2018) 502-518

Abstract

Let ψ:R+→R+ψ:\mathbb R_+\to\mathbb R_+ be a non-increasing function. A real number xx is said to be ψψ-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system ∣qx−p∣< ψ(t)  and  ∣q∣<t|qx-p|< \, ψ(t) \ \ {\text{and}} \ \ |q|<t has a non-trivial integer solution for all large enough tt. Denote the collection of such points by D(ψ)D(ψ). In this paper, we prove that the Hausdorff measure of the complement D(ψ)cD(ψ)^c (the set of ψψ-Dirichlet non-improvable numbers) obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

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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.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements3 reported findingsContains wrong statements

The printed argument for the zero-infinity laws does not cover every auxiliary function allowed by Theorems 1.4 and 1.6, and the unconditional Hausdorff-dimension formula is false for functions allowed by the printed hypotheses.

Dimension formula after Theorem 1.4Incorrect

The displayed dimension formula is false in the stated generality

Page 3 · consequence following Theorem 1.4 · arXiv:1704.03089v1

Take the allowed function ψ(t)=t−2\psi(t)=t^{-2}. Then Ψ(t)=tψ(t)/(1−tψ(t))=1/(t−1)\Psi(t)=t\psi(t)/(1-t\psi(t))=1/(t-1) and τ=−1\tau=-1, so the printed formula gives 2/(2+τ)=22/(2+\tau)=2. In fact D(ψ)=QD(\psi)=\mathbb{Q}. Every rational belongs to D(ψ)D(\psi) by using a fixed exact denominator. If xx is irrational and qnq_n are its convergent denominators, Lagrange optimality gives ∣qx−p∣≥∣qn−1x−pn−1∣=1/(qn+Tn(x)qn−1)>1/(2qn)>qn−2|qx-p|\geq|q_{n-1}x-p_{n-1}|=1/(q_n+T^n(x)q_{n-1})>1/(2q_n)>q_n^{-2} for every 0<∣q∣<qn0<|q|<q_n once qn>2q_n>2; the defining system therefore fails at t=qnt=q_n. Hence D(ψ)cD(\psi)^c has Hausdorff dimension one. Theorem 1.4's own series also diverges for every s<1s<1. Restricting the displayed consequence to τ≥0\tau\geq0 avoids this counterexample; a formula outside that regime must be stated separately.

Full paper, version 1 ↗
Theorems 1.4 and 1.6Not able to verify

The proof does not cover every auxiliary function permitted by the statements

Pages 3–5 · Theorems 1.4, 1.6, and 1.8 · arXiv:1704.03089v1

Theorems 1.4 and 1.6 assume only that the original approximating function is non-increasing. Their stated deduction is from Theorem 1.8, which explicitly requires the associated function Ψ\Psi to be non-decreasing, and that property does not follow from the printed hypotheses. The paper supplies no monotone-envelope argument for the omitted cases.

Continued-fraction equivalenceTypo · no additional status impact

One introductory inequality sign is reversed

Page 4 · display preceding the continued-fraction criterion · arXiv:1704.03089v1

The introductory display uses less-than for the product of the two continued-fraction tails, whereas the later lemma and its direct algebra give greater-than. Reversing that one sign matches the proof and all subsequent inclusions.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The proof of the monotone auxiliary-function theorem is complete, but the claimed deduction of the broader main theorems omits a necessary monotonicity bridge and cannot support the false dimension formula.

Proof of Theorem 1.8Correct and complete under the stated monotonicity hypothesis

The zero-infinity argument is correct under its own hypotheses

Pages 8–15 · proof of Theorem 1.8 · arXiv:1704.03089v1

The divergence half follows from the continued-fraction limsup subset and the Jarnik-type theorem, while the convergence half uses a cylinder cover whose double sum is controlled by essential sublinearity. Monotonicity of Ψ\Psi is used in both reductions.

Deduction of Theorems 1.4 and 1.6Incomplete as written; no repair supplied

A required monotonicity step is missing

Page 5 · sentence deducing the main theorem from Theorem 1.8 · arXiv:1704.03089v1

The deduction invokes Theorem 1.8 without proving that Ψ\Psi is non-decreasing. The allowed example ψ(t)=t−2\psi(t)=t^{-2} makes Ψ(t)=1/(t−1)\Psi(t)=1/(t-1) strictly decreasing, so this is a genuine gap in scope rather than an automatic consequence of the definitions.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1704.03089v1
Authors listed
Mumtaz Hussain, Dmitry Kleinbock, Nick Wadleigh, Bao-Wei Wang
Audit date
August 19, 2026
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