arXiv:0905.0830v1

Badly approximable numbers and Littlewood-type problems

Yann Bugeaud, Nikolay Moshchevitin

math.NT11J1311J2511K60

Paper metadata

Original submission: May 6, 2009
Claimed MSC: 11J13, 11J25, 11K60
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 14 pages

Abstract

We establish that the set of pairs $(α, β)$ of real numbers such that $\liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q α\Vert \cdot \Vert q β\Vert > 0,$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$ . Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems

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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 20, 2026

01Statements4 reported findingsCorrect

The full-dimension results for multiplicatively badly approximable pairs, their logarithmically refined and mixed $p$ -adic forms, and the dual two-variable form are correct. Theorem 4 needs only the displayed boundary correction $x,y\geq1$ : allowing a zero coordinate would trivially force its infimum to vanish. Its two-dimensional Peres–Schlag proof also needs a local geometric repair, supplied below, which preserves the theorem. No substantive counterexample to a central result was found.

Theorems 1 and 2Correct

Multiplicatively badly approximable pairs have full dimension

Pages 2 and 7–10 · Theorems 1–2 and proof of Theorem 1 · arXiv:0905.0830v1

For each fixed $\alpha\in\operatorname{Bad}$ , the dangerous intervals around $y/x$ have radii proportional to $(\lVert\alpha x\rVert x^2(\log x)^2)^{-1}$ . Lemma 2 makes their total loss uniformly small on every cubic block. The nested dyadic construction therefore retains a Cantor set, and its branching/separation ratio tends to one, giving full dimension in the $\beta$ fiber. Since the set of admissible $\alpha$ also has dimension one, the cited slicing principle gives dimension two for the set of pairs.

Theorem 3Correct

The logarithmically refined fiber theorem

Page 3 and pages 6 and 10 · Theorem 3, Lemma 3, and proof reduction · arXiv:0905.0830v1

Bounded continued-fraction quotients for $\alpha\in\operatorname{Bad}$ give uniform distribution on each denominator block and $\sum_{x=q}^{q^3}\frac{1}{\lVert\alpha x\rVert x(\log(1/\lVert\alpha x\rVert))^a}\ll(\log q)^{2-a}.$ This is exactly the loss estimate needed after replacing $(\log x)^2$ by $(\log x)^{2-a}(\log(1/\lVert\alpha x\rVert))^a$ . The original branching argument is unchanged, so the full-dimension conclusion follows.

Theorem 5 and Corollary 1Correct

The mixed $p$ -adic full-dimension result

Pages 4 and 7 and 10 · Theorem 5, Lemma 4, and proof reduction · arXiv:0905.0830v1

Splitting integers by $v_p(x)=j$ yields $\sum_{x=q}^{q^3}\frac{1}{x|x|_p(\log(2/|x|_p))^a}\ll(\log q)^{2-a}.$ Thus the dangerous intervals with the mixed $p$ -adic weight satisfy the same block-loss bound as in Theorem 1. The Cantor construction gives full Hausdorff dimension, and setting $a=0$ gives Corollary 1.

Theorem 4Minor formal correction

The dual two-variable theorem needs a boundary correction

Page 4 and pages 10–11 · Theorem 4 and proof outline · arXiv:0905.0830v1

The displayed infimum must range over positive integers $x,y\geq1$ , or equivalently over integer pairs with $xy\neq0$ . With the printed $x,y\geq0$ , setting $x=0$ and taking denominators of $\beta$ forces $2\lVert y\beta\rVert(\log2)^2$ to have infimum zero. Every subsequent estimate groups pairs by $q\leq|xy|\leq q^3$ and therefore already uses $xy\neq0$ . Changing the boundary in the one displayed statement repairs the issue without altering any related claim.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The one-dimensional Cantor construction and the logarithmic and $p$ -adic adaptations are correct. The proof outline for Theorem 4 incorrectly says that a fixed pair $(x,y)$ determines a unique integer level in the unit square and omits the resulting local strip-count calculation. Accounting for all relevant levels supplies a verified repair with the same summability bound.

Proof of Theorem 1Correct

The dyadic survival and mass-distribution estimates close

Pages 7–10 · equations (4.1)–(4.10) and conclusion · arXiv:0905.0830v1

Bad approximability supplies both $\lVert q\alpha\rVert\geq\delta/q$ and the uniform block sum in Lemma 2. Each retained dyadic interval loses at most a fixed small fraction to the next cubic block of dangerous sets. The construction keeps at least $m_k=2^{L_{k+1}-L_k-5}$ mutually separated children. Since $2^{L_k}\gg q_k^2(\log q_k)^2$ and $q_k=q_0^{3^k}$ , the mass-distribution quotient tends to one. Hence the resulting set is nonempty and has Hausdorff dimension one.

Lemmas 2–4Correct

All three block-sum estimates have the required uniform scale

Pages 5–7 · Lemmas 2–4 · arXiv:0905.0830v1

Lemma 2 follows by dividing the classical partial sum $\sum_{x\leq m}(\lVert\alpha x\rVert x)^{-1}\ll(\log m)^2$ by the minimum $(\log x)^2$ on $[q,q^3]$ . Lemma 3 decomposes into convergent-denominator blocks; bounded partial quotients make their number $O(\log q)$ and each weighted block $O((\log q)^{1-a})$ . Lemma 4 decomposes by the exact $p$ -adic valuation and sums $O(\log q/(j+1)^a)$ over $j=O(\log q)$ . These are precisely the uniform bounds used later.

Proof outline for Theorem 4Incorrect as written · verified repair

A fixed normal vector has many integer levels, not one

Pages 10–11 · proof of Theorem 4 · arXiv:0905.0830v1

For fixed positive integers $(x,y)$ , the unit square meets generally $x+y+O(1)$ lines $xX+yY+z=0$ , not a unique line as stated. Each corresponding dangerous strip has width $O(\varepsilon_{x,y}/\sqrt{x^2+y^2})$ and intersection length at most $\sqrt2$ . Since $(x+y)/\sqrt{x^2+y^2}=O(1)$ , the total area over all relevant $z$ is still $O(\varepsilon_{x,y})$ . The same estimate holds locally in every retained dyadic square, with only a boundary constant. Therefore $\sum_{q\leq xy\leq q^3}O(\varepsilon_{x,y})\ll\frac{\varepsilon}{\log q},$ and the two-dimensional Cantor/mass-distribution construction proceeds exactly as in Theorem 1. This supplies the missing multiplicity calculation and verifies the theorem.

Proof of Theorem 3Typo

The convergent-index order is reversed in one sentence

Page 6 · proof of Lemma 3 · arXiv:0905.0830v1

If $m$ is the largest index with $q_m\leq q$ and $n$ the smallest with $q_n\geq q^3$ , then $m\leq n$ and both are comparable to $\log q$ . The printed chain $\log q\ll n\ll m\ll\log q$ reverses their order; it should state $\log q\ll m\leq n\ll\log q$ . The proof immediately sums from $j=m$ to $n$ and therefore uses the corrected order.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:0905.0830v1
Authors listed: Yann Bugeaud, Nikolay Moshchevitin
Audit date: August 20, 2026

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Badly approximable numbers and Littlewood-type problems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Multiplicatively badly approximable pairs have full dimension

The logarithmically refined fiber theorem

The mixed ppp-adic full-dimension result

The dual two-variable theorem needs a boundary correction

The dyadic survival and mass-distribution estimates close

All three block-sum estimates have the required uniform scale

A fixed normal vector has many integer levels, not one

The convergent-index order is reversed in one sentence

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The mixed $p$ -adic full-dimension result