arXiv:1405.7650v6
Intrinsic Diophantine approximation on quadric hypersurfaces
math.NTmath.DS11K6037A17
Abstract
We consider the question of how well points in a quadric hypersurface can be approximated by rational points of . This contrasts with the more common setup of approximating points in a manifold by all rational points in . We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani ('86) and Kleinbock--Margulis ('99).
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01Statements2 reported findingsCorrect
The intrinsic Dirichlet theorem by rational rank, full-dimension theorem for intrinsically badly approximable points, and Khintchine-type theorem on nonsingular rational quadrics are correct.
Theorem 5.1Correct
Intrinsic Dirichlet theorem for rational quadrics
Section 5 · Theorem 5.1 · arXiv:1405.7650v6
The correspondence sends rational points on the quadric to primitive isotropic vectors and approximation quality to cusp depth. Reduction theory supplies such a vector at every sufficiently large height with the exponent dictated by the rational rank, including the separate anisotropic cases stated in the theorem.
Full paper, version 6 ↗Theorems 4.5 and 6.3Correct
Badly approximable dimension and Khintchine law
Sections 4 and 6–8 · Theorems 4.5 and 6.3 · arXiv:1405.7650v6
Bounded trajectories in the orthogonal homogeneous space characterize intrinsic bad approximation, and the winning construction gives full dimension. For the metric theorem, reduction theory identifies the cusp tail, and the exponentially mixing shrinking-target theorem converts its measure asymptotics into the displayed convergence-divergence law, with the exceptional low-rank case treated separately.
02Proofs2 reported findingsCorrect
The intrinsic dynamical correspondence, reduction-theoretic cusp estimates, winning argument, and shrinking-target proof are correct and complete.
Sections 3–5Correct and complete
Rational approximation is equivalent to orthogonal cusp excursions
Sections 3–5 · Corollary 4.3 and Theorems 4.5, 5.1 · arXiv:1405.7650v6
The chosen normalization identifies projective height with the relevant lattice-vector coordinate up to uniform constants. Mahler compactness turns a lower vector bound into boundedness, and the rank cases in the reduction argument give exactly the stated Dirichlet functions.
Sections 7–8Correct and complete
Cusp tails satisfy the Borel–Cantelli hypotheses
Sections 7–8 · Theorem 7.4 and Proposition 8.9 · arXiv:1405.7650v6
The height function is distance-like, its tail has the computed exponential order, and the diagonal flow is exponentially mixing on each relevant homogeneous component. Substitution of the approximation function into this tail integral gives the series in Theorem 6.3, including the explicitly isolated exceptional form.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.