arXiv:math/0506510v1
Flows on -arithmetic homogeneous spaces and applications to metric Diophantine approximation
math.NTmath.DS11J8337A13
Abstract
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and -adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of ) are generalized to the -arithmetic setting.
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01Statements2 reported findingsCorrect
The quantitative nondivergence theorem over products of local fields, its uniform homogeneous-space recurrence consequence, and the resulting strong-extremality theorem for nondegenerate -arithmetic maps are correct under the stated Federer, goodness, and nonplanarity hypotheses.
Theorem 8.3Correct
Quantitative -arithmetic nondivergence
Section 8 · Theorem 8.3 · arXiv:math/0506510v1
The theorem follows after assigning every primitive submodule its covolume norm: those functions satisfy the norm-like axioms, the stated -goodness condition supplies the small-value estimate, and the lower bound by permits the poset induction. The resulting exceptional-set bound has the announced exponent and applies simultaneously at all places.
Full paper, version 1 ↗Theorems 9.1 and 10.4Correct
Uniform recurrence and strong extremality
Sections 9–12 · Theorems 9.1 and 10.4 · arXiv:math/0506510v1
The linearization alternative in Theorem 9.1 gives either uniform compact recurrence or a rational-subgroup obstruction. For a good nonplanar map the obstruction is excluded by the exterior-algebra lower bounds, while the Dani correspondence converts recurrence estimates into convergence of the relevant multiplicative Diophantine limsup sets. This proves the stated almost-everywhere strong extremality conclusion.
02Proofs2 reported findingsCorrect
The local-field good-function estimates, covolume induction, linearization argument, and dynamical-to-Diophantine correspondence are compatible and cover the archimedean and ultrametric factors.
Propositions 3.3 and Theorem 8.3Correct and complete
Good functions feed the nondivergence induction
Sections 3 and 8 · arXiv:math/0506510v1
Polynomial and analytic coordinate combinations have the required uniform good-function bounds on product balls. Ultrametric balls supply the covering property used in the induction, and primitive submodules form the discrete partially ordered family required by the quantitative nondivergence theorem.
Sections 9–12Correct and complete
The recurrence alternative yields the metric theorem
Sections 9–12 · arXiv:math/0506510v1
Compactness is expressed by lower bounds for all primitive-submodule covolumes. The linearization step transfers failure of recurrence to an algebraic relation; nonplanarity rules that relation out. The final Borel–Cantelli application uses the same height and local norms as the stated multiplicative approximation problem, so no parameter regime is lost.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.