arXiv:1507.07119v2

Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Paloma Bengoechea, Nikolay Moshchevitin

math.NT

Paper metadata

Original submission: July 25, 2015
Latest revision: July 25, 2016 · version 2
Claimed MSC
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗

Abstract

For any $j_1,\ldots,j_n>0$ with $j_1+\cdots+j_n=1$ and any $x\in\mathbb{R}^n$ , we consider the set of points $y\in\mathbb{R}^n$ for which $\max_{1\leq i\leq n}\lVert qx_i-y_i\rVert^{1/j_i}>c/q$ for some positive constant $c=c(y)$ and all $q\in\mathbb{N}$ . These sets are the twisted inhomogeneous analogue of $\operatorname{Bad}(j_1,\ldots,j_n)$ in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e. when $j_i=1/n$ , and in the weighted setting when $x$ is chosen from $\operatorname{Bad}(j_1,\ldots,j_n)$ . We generalise these results, proving full Hausdorff dimension in the weighted setting without any condition on $x$ .

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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 20, 2026

01Statements2 reported findingsCorrect

The full Hausdorff-dimension theorem for weighted twisted badly approximable points, including simultaneous coordinatewise badness, is supported for every twist vector.

Theorem 1.1Correct

The intersection has full ambient Hausdorff dimension

Sections 1 and 4-6 · Theorem 1.1 and Cantor construction · arXiv:1507.07119v2

For each large subdivision parameter $R$ , the construction removes hyperrectangles meeting either a dual best-approximation resonance or a coordinate rational resonance. Each weighted parent has $R$ children, and the paper's exact lower count is $R-\sum_i R^{j_i}-O((1+\log R)R^{1-j_{\min}})$ , which is positive for large $R$ . The resulting mass estimate tends to exponent $n$ as $R\to\infty$ , proving the stated dimension.

Proposition 3.1Correct

Dual avoidance implies twisted bad approximation

Section 3 · Proposition 3.1 and inequalities (8)-(11) · arXiv:1507.07119v2

Choosing consecutive dual best approximations so that $q$ lies between their reciprocal errors makes the term $q\zeta_\nu$ small. Minkowski's inequality $\zeta_\nu M_{\nu+1}\leq1$ then converts the fixed lower bound on $\|m_\nu\cdot\eta\|$ into a uniform positive lower bound for $\max_i q^{j_i}\|q\theta_i-\eta_i\|$ .

02Proofs2 reported findingsCorrect

The best-approximation, removal-count, and mass-distribution steps consistently yield the claimed dimension without an unproved case split.

Lemma 2.2Correct

The weighted best-approximation heights are quantitatively lacunary

Section 2 · Lemma 2.2 · arXiv:1507.07119v2

A doubled weighted approximation body is covered by $3^n$ translates of the preceding body. The best-approximation minimality bounds the lattice points in those translates and yields $M_{\nu+2\cdot3^n}\geq2M_\nu$ , which is the exact finite-per-scale estimate later used.

Sections 4-6Correct and complete

The Cantor measure supplies the full lower dimension

Sections 4-6 · child counts, Lemma 5.1, and final dimension bound · arXiv:1507.07119v2

The hyperplane and coordinate resonances intersect only the explicitly counted children. Assigning equal mass to survivors gives $\mu(S)\ll |S|^{s(R)}$ for sufficiently small cubes, where $s(R)\to n$ . The mass distribution principle therefore gives dimension at least $n$ , matching the ambient upper bound.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:1507.07119v2
Authors listed: Paloma Bengoechea, Nikolay Moshchevitin
Audit date: August 20, 2026

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Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The intersection has full ambient Hausdorff dimension

Dual avoidance implies twisted bad approximation

The weighted best-approximation heights are quantitatively lacunary

The Cantor measure supplies the full lower dimension

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