Non-dense orbits on homogeneous spaces and applications to geometry and number theory

An, Jinpeng; Guan, Lifan; Kleinbock, Dmitry

doi:10.1017/etds.2021.4

Cited by 7 publications

(11 citation statements)

References 29 publications

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“…In order to prove the thickness result for the Euclidean norm, we use a result of the first-named author with An and Guan [2]. They give a very general condition on the critical locus L ν which guarantees that the set of A ∈ M m,n such that the trajectory {a s hx : s > 0} eventually stays away from L ν is winning in the sense of Schmidt.…”

Section: Thickness Results Via Transversalitymentioning

confidence: 99%

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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Section: Thickness Results Via Transversalitymentioning

confidence: 99%

Weighted uniform Diophantine approximation of systems of linear forms

Kleinbock

Rao

2022

Pure and Applied Mathematics Quarterly

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“…More precisely, the results in [AGK] deal with a modified version of Schmidt's winning property called hyperplane absolute winning (HAW). For the definition of the HAW property, see [BFKRW,§2] or [AGK,§2.1]. HAW implies winning in the sense of Schmidt [Sc1], and this in turn implies thickness.…”

Section: Thickness Results Via Transversalitymentioning

confidence: 99%

Weighted uniform Diophantine approximation of systems of linear forms

Kleinbock¹,

Rao²

2021

Preprint

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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 SLm+n(R)/ SLm+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 one-parameter 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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“…To reconcile our notation with that of [AGK,Theorem 2.8], we point out that for H and F as in (4.1), H is actually the maximally expanding horospherical subgroup of SL 3 (R) with respect to F + := {a s : s ≥ 0}, as defined in [AGK,§2.3]. Hence H coincides with the subgroup H max F + , and, consequently, condition (2.16) of [AGK,Theorem 2.8] is equivalent to the (F, H)-transversality. Also F is diagonalizable, therefore condition (2.17) is redundant.…”

Section: The Interaction Of the Critical Locus And The Flowmentioning

confidence: 99%

“…Also F is diagonalizable, therefore condition (2.17) is redundant. This is explained in [AGK,Remark 2.9(2)].…”

Section: The Interaction Of the Critical Locus And The Flowmentioning

confidence: 99%

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Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

Kleinbock,

Rao

2021

Preprint

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In the paper [AS] of Akhunzhanov-Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on R 2 and prove that, for each such norm, the set of Dirichlet improvable pairs has the hyperplane absolute winning property. The method of proof is an orbit avoidance theorem in the space of lattices due to the first-named author with An and Guan, and some classical results in the geometry of numbers dating back to Chalk-Rogers and Mahler. As a corollary, we conclude that for any norm on R 2 the top of the Dirichlet spectrum is not an isolated point.

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Non-dense orbits on homogeneous spaces and applications to geometry and number theory

Cited by 7 publications

References 29 publications

Weighted uniform Diophantine approximation of systems of linear forms

Weighted uniform Diophantine approximation of systems of linear forms

Weighted uniform Diophantine approximation of systems of linear forms

Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

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