arXiv:1405.5545v2

On the complexity of a putative counterexample to the $p$ -adic Littlewood conjecture

Dmitry Badziahin, Yann Bugeaud, Manfred Einsiedler, Dmitry Kleinbock

math.NTmath.DS11J0411J6111J8337A3537A4537D40

Paper metadata

Original submission: May 21, 2014
Latest revision: January 22, 2015 · version 2
Claimed MSC: 11J04, 11J61, 11J83, 37A35, 37A45, 37D40
arXiv categories: math.NT, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 19 pages; a revised version
Journal reference: Compositio Mathematica 151 (2015) 1647-1662
DOI: 10.1112/S0010437X15007393 ↗

Abstract

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$ , let $| \cdot |_p$ denote the $p$ -adic absolute value. In 2004, de Mathan and Teulié asked whether $\inf_{q \ge 1} \, q \cdot || q α|| \cdot | q |_p = 0$ holds for every badly approximable real number $α$ and every prime number $p$ . Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $α$ grows too rapidly or too slowly, then their conjecture is true for the pair $(α, p)$ with $p$ an arbitrary prime.

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Audited against arXiv v2

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 lower-complexity obstructions to a putative counterexample to the $p$ -adic Littlewood conjecture, the quantitative complexity bound, and the Lagrange-constant consequences are correct.

Theorems 2.1 and 2.3Correct

Complexity and recurrence restrictions

Section 2 · Theorems 2.1 and 2.3 · arXiv:1405.5545v2

The continued-fraction coding converts repeated blocks into near returns of the associated lattice orbit. Measure rigidity forces any orbit closure supporting the required recurrent pattern either to meet the forbidden locus or to have the high symbolic complexity stated in the theorem.

Full paper, version 2 ↗

Theorem 2.7 and Theorem 3.3Correct

Quantitative and Lagrange-spectrum consequences

Sections 2–3 · Theorems 2.7 and 3.3 · arXiv:1405.5545v2

Counting distinct length- $n$ blocks supplies the announced quantitative lower bound after applying the recurrence alternative. The same coding identifies the relevant $p$ -adic Lagrange constants with orbit-closure minima, yielding the alternatives in Theorem 3.3.

02Proofs2 reported findingsCorrect

The symbolic coding, recurrence estimates, and measure-rigidity applications are correct and complete.

Section 2Correct and complete

Repeated blocks give the required homogeneous-space returns

Section 2 · proofs of Theorems 2.1–2.7 · arXiv:1405.5545v2

Continuant identities control the lattice coordinates at both ends of a repeated word. Low block complexity forces sufficiently many controlled returns, while the assumed counterexample keeps the orbit away from the zero-product locus; applying the rigidity alternative yields the claimed contradiction and quantitative bound.

Section 3Correct and complete

The Lagrange constants follow from the same orbit correspondence

Section 3 · Theorem 3.3 · arXiv:1405.5545v2

The liminf defining each constant is the infimum of the corresponding height over the orbit closure. Compactness and invariance justify passage to the closure, and the dichotomy from Section 2 applies with no change in the normalization.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:1405.5545v2
Authors listed: Dmitry Badziahin, Yann Bugeaud, Manfred Einsiedler, Dmitry Kleinbock
Audit date: August 19, 2026

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On the complexity of a putative counterexample to the ppp-adic Littlewood conjecture

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Complexity and recurrence restrictions

Quantitative and Lagrange-spectrum consequences

Repeated blocks give the required homogeneous-space returns

The Lagrange constants follow from the same orbit correspondence

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On the complexity of a putative counterexample to the $p$ -adic Littlewood conjecture