Nondense orbits on homogeneous spaces and applications to geometry and number theory

An, Jinpeng; Guan, Lifan; Kleinbock, Dmitry

doi:10.48550/arxiv.2001.05174

Cited by 3 publications

(1 citation statement)

References 33 publications

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“…The Hausdorff dimensions of such sets are intensively investigated, which sometimes led to interesting results in number theory and other fields. For example, see [10,11,12,22,21,6,23,2,19,1,3]. Similar results are also established for more general hyperbolic or partially hyperbolic systems [33,9,14,20,32,35,36,37].…”

Section: Introductionmentioning

confidence: 53%

Non-dense orbits of systems with approximate product property

Sun¹

2019

Preprint

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Section: Introductionmentioning

confidence: 53%

Non-dense orbits of systems with approximate product property

Sun¹

2019

Preprint

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Weighted uniform Diophantine approximation of systems of linear forms

Kleinbock

Rao

2022

Pure and Applied Mathematics Quarterly

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Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an m × n matrix are governed by a trajectory in SL m+n (R)/ SL m+n (Z) avoiding a compact subset of the space of lattices called the critical locus defined with respect to the corresponding norm. The trajectory is formed by the action of a oneparameter diagonal subgroup corresponding to the weights. We first state a very precise form of Dirichlet's theorem and prove it for some norms. Secondly we show, for these same norms, that the set of Dirichlet-improvable matrices has full Hausdorff dimension. Though the techniques used vary greatly depending on the chosen norm, we expect these results to hold in general.

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A zero-one law for uniform Diophantine approximation in Euclidean norm

Kleinbock

Rao

2019

Preprint

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We state a norm sensitive Diophantine approximation problem arising form the work of the first-named author and Wadleigh [KWa], and extend the set-up by replacing the supremum norm with an arbitrary norm, following the approach of Andersen and Duke [AD]. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices. In particular, the choice of the Euclidean norm on R 2 corresponds to studying geodesics on a hyperbolic surface which visit a decreasing family of balls. An application of the dynamical Borel-Cantelli lemma of Maucourant [Mau] produces, given an approximation function ψ, a zero-one law for the set of α ∈ R for which the inequality αq−p ψ(t) 2 + q t 2 < 2 √ 3 has non-trivial integer solutions for all large enough t.

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Nondense orbits on homogeneous spaces and applications to geometry and number theory

Cited by 3 publications

References 33 publications

Non-dense orbits of systems with approximate product property

Non-dense orbits of systems with approximate product property

Weighted uniform Diophantine approximation of systems of linear forms

A zero-one law for uniform Diophantine approximation in Euclidean norm

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