Published paper

Diophantine approximation and badly approximable sets

Simon Kristensen, Rebecca Thorn, Sanju Velani

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Publication metadata

Journal citation: Advances in Mathematics 203(1), 132–169 (2006)
Publication date: 2006
DOI: 10.1016/j.aim.2005.04.005 ↗
arXiv counterpart: arXiv:math/0405433v3

Role in dependence graphs

Proof-critical source

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

This paper is included only for the following marked statement:

Theorem 1 and Lemma 7 · arXiv:math/0405433v3The general full-dimension criterion and local counting estimate used by BHKV.

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Frozen exact arXiv:math/0405433v3 · final-manuscript fallback

Simon Kristensen, Rebecca Thorn, and Sanju Velani. Diophantine approximation and badly approximable sets. arXiv:math/0405433v3; Advances in Mathematics 203 (2006), 132–169.

The publisher landing page was available but the exact version-of-record full text was not acquired. The report therefore evaluates the authors' exact final arXiv v3 manuscript and does not claim to audit the version of record.

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Generated September 26, 2026

01Statements2 reported findingsNo statement issues found

The abstract full-dimension criteria and the advertised number-theoretic and dynamical applications are correct after a verified logical-quantifier repair in the rectangular framework. The ball-case cardinality hypothesis is a lower bound, as printed in Theorem 1.

Theorem 1No issue found

The ball framework has the correct cardinality direction and yields full dimension

arXiv:math/0405433v3 · §1.3, Theorem 1; proof through §§3–4

The retained-child hypothesis is correctly printed as the lower bound $\#\mathcal C(\theta B_n)\geq\kappa_1(\rho(k^n)/\rho(k^{n+1}))^\delta$ . Removing at most the corresponding $\kappa_2$ proportion leaves a uniformly branching Cantor construction because $\kappa_2<\kappa_1$ . The constructed probability measure satisfies the required Frostman estimate, so the mass distribution principle gives dimension tending to $\delta$ as $k\to\infty$ .

Exact final arXiv v3 manuscript ↗

ApplicationsNo issue found

The simplex and separation estimates verify the abstract hypotheses in the main examples

arXiv:math/0405433v3 · §5, including the simplex lemma and the real, complex, $p$ -adic, formal-series, IFS, rational-map, and Kleinian-group applications

In the simultaneous real case, rational points with comparable denominators in a sufficiently small convex region lie on one affine hyperplane by the determinant lower bound. Counting the next-scale rectangles meeting that hyperplane gives the required bad-child estimate. The later applications use the analogous separation or distortion estimates appropriate to their ambient spaces, with the measure-growth exponents substituted consistently into the abstract theorem.

02Proofs3 reported findingsSuspected incomplete or incorrect proofs

The Cantor-measure proof and application-specific counting arguments are sound, but the rectangular-framework proof repeatedly replaces avoidance of a product neighborhood by coordinatewise avoidance in every coordinate. That equivalence is false as written. A verified repair using an existential coordinate at each resonant set preserves every theorem and application.

Rectangular avoidance quantifierSuspected incorrect as written · an apparently sound repair

Leaving a product neighborhood requires one separated coordinate, not all coordinates

arXiv:math/0405433v3 · §2.1 after the definition of $\operatorname{Bad}^*$ ; §4, condition (C), induction, and completion of proof

For a product neighborhood $F_\alpha=\{x:d_i(x_i,R_{\alpha,i})\leq\rho_i(\beta_\alpha)\text{ for all }i\}$ , the condition $x\notin cF_\alpha$ means that for each $\alpha$ there exists at least one coordinate $i=i(\alpha)$ with $d_i(x_i,R_{\alpha,i})>c\rho_i(\beta_\alpha)$ . The manuscript instead states several times that the inequality holds for every coordinate. Repair each such displayed all-coordinate claim by the logically equivalent maximum condition $\max_i\frac{d_i(x_i,R_{\alpha,i})}{\rho_i(\beta_\alpha)}\geq c.$ The bad-child hypothesis already counts rectangles simultaneously close in all coordinates, so every retained child has the repaired property. For the finitely many small-height resonant sets, compactness and exclusion of their union give a positive minimum of the same maximum. The induction, dimension estimate, and all applications then proceed unchanged.

Exact final arXiv v3 manuscript ↗

Cantor measureCorrect and complete after an apparently sound repair

The lower-dimension estimate is complete after the quantifier repair

arXiv:math/0405433v3 · §§3–4, construction of $\mathbf K_{c(k)}$ and the measure $\mu$

The retained rectangles have uniform branching bounded below by the difference $\kappa_1-\kappa_2$ . Equal splitting defines a probability measure. A ball at an intermediate scale meets only a controlled number of children, while the doubling and comparability hypotheses convert this count into $\mu(B)\ll r^{\delta-2\varepsilon(k)}$ . Since $\varepsilon(k)\to0$ , the mass distribution principle yields the claimed lower bound.

Mass distribution inputCorrectly sourced terminal input

The cited terminal principle is applied with the correct inequality

arXiv:math/0405433v3 · §3, Mass Distribution Principle; applied in §4

The source cites Falconer's Fractal Geometry, page 55, for the standard implication from a probability measure satisfying $\mu(B)\leq c r^s$ at small scales to $\dim\Omega\geq s$ . The constructed measure meets exactly this hypothesis.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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