S-adic version of Minkowski's geometry of numbers and Mahler's compactness criterion

Kleinbock, Dmitry; Shi, Ronggang; Tomanov, George

doi:10.1016/j.jnt.2016.10.016

Cited by 14 publications

(9 citation statements)

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“…Using Theorem 1.2 of [KST17] we get that ∃ s > 1 such that λ 1 (Λ) ≤ s − 1 for all Λ ∈ X. Therefore for any vector #» v 1 ∈ Λ with #» v 1 2 = λ 1 (Λ), any annular region of width s would contain some integer multiple of #» v 1 .…”

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confidence: 96%

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

Alam

Ghosh

2020

Trans. Amer. Math. Soc.

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The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss [KSW17] regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M. Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove "spiraling" results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh and Tseng [AGT15, AGT14] as well as Kleinbock, Shi and Weiss [KSW17].If · is taken to be the supremum norm then c m can be taken to be 1. In [AGT15], Athreya, Ghosh and Tseng considered the problem of 'spiraling' of approximates connected to the Diophantine inequality above. LetGiven A ⊆ S m−1 , T > 0, they considered the counting functions N(x, T ) = #{(p, q) ∈ Z m × Z + : qx − p < c m |q| −1/m , 0 < q ≤ T } and N(x, T, A) = #{(p, q) ∈ Z m × Z + : qx − p < c m |q| −1/m , 0 < q ≤ T, θ(p, q) ∈ A}, and proved: A.

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confidence: 96%

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

Alam

Ghosh

2020

Trans. Amer. Math. Soc.

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“…Let Q be a compact subset of X . By Mahler's compactness criterion (see for instance [KlST,Theo. 1.1]), there exists ε P s0, 1r such that Q is contained in the ǫ-thick part…”

Section: Dani Correspondencementioning

confidence: 99%

“…By Mahler's compactness criterion (see for instance [KlST,Theo. 1.1]) and since the natural projection π : Y Ñ X is proper, the subset L ε is compact.…”

Section: Dani Correspondencementioning

confidence: 99%

“…The proofs of the above main theorems of this paper are largely divided into two parts. Firstly, assuming the singular on average property in order to prove the full Hausdorff dimension property, we give a lower bound on the Hausdorff dimension of appropriately chosen subsets of K m v , using new function fields versions of classical tools in Diophantine approximation such as geometry of numbers, transference principle and best approximation vectors (see for instance [Cas,Sch3,Kri,KlW,Cheu,Chev,GE,CC,KlST,GG,BZ,Ger,LSST,CGGMS,BuKLR]). Secondly, in order to prove the upper bound in Theorem 1.2, we use technics of homogeneous dynamics of diagonal actions and in particular the entropy method (see for instance [Kle, EL, LSS, ELW]).…”

Section: Introductionmentioning

confidence: 99%

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On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

Kim¹,

Lim²,

Paulin³

2021

Preprint

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In this paper, we study inhomogeneous Diophantine approximation over the completion K v of a global function field K (over a finite field) for a discrete valuation v, with affine algebra R v . We obtain an effective upper bound for the Hausdorff dimension of the set}q} n }Aq ´θ ´p} m ě ǫ * , of ǫ-badly approximable targets θ P K m v for a fixed matrix A P M m,n pK v q, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of R v -grids. We further characterize matrices A for which Bad A pǫq has full Hausdorff dimension for some ǫ ą 0 by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances. 1 2 Background material for the lower bound On global function fieldsWe refer for instance to [Gos, Ros], as well as [BPP, §14.2], for the content of this section. Let F q be a finite field with q elements, where q is a positive power of a positive prime. Let K be the function field of a geometrically connected smooth projective curve C over F q , or equivalently an extension of F q of transcendence degree 1, in which F q is algebraically closed. We denote by g the genus of C. There is a bijection between the set of closed points of C and the set of normalized discrete valuations v of K, the valuation of a given element f P K being the order of the zero or the opposite of the order of the pole of f at the given closed point. We fix such an element v throughout this paper, and use the notation K v , O v , π v , k v , q v , | ¨| defined in the introduction. We furthermore denote by deg v the degree of v, so that q v " q deg v .We denote by vol v the normalized Haar measure on the locally compact additive group K v such that vol v pO v q " 1. For any positive integer d, let vol d v be the normalized Haar measure on K d v such that vol d v pO d v q " 1. Note that for every g P GL d pK v q we have

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“…The adelic successive minima theorem has been established independently by McFeat [40] and Bombieri and Vaaler [7]. See also [9,30], and the references therein for S-adic version of other foundational results of geometry of numbers. There seem to exist multiple versions, although with very little variance, of Dirichlet's theorem over number fields in the literature ( [9,24,43,44,45]).…”

Section: Introductionmentioning

confidence: 99%

Dirichlet improvability for $S$-numbers

Das¹,

Ganguly²

2022

Preprint

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We study the problem of improving Dirichlet's theorem of metric Diophantine approximation in the S-adic setting. Our approach is based on translation of the problem related to Dirichlet improvability into a dynamical one, and the main technique of our proof is the S-adic version of quantitative nondivergence estimate due to D. Y. Kleinbock and G. Tomanov. The main result of this paper can be regarded as the number field version of earlier works of D. Y. Kleinbock and B. Weiss, and of the second named author and Anish Ghosh.

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S-adic version of Minkowski's geometry of numbers and Mahler's compactness criterion

Abstract: Abstract. In this note we give a detailed proof of certain results on geometry of numbers in the S-adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.

Cited by 14 publications

References 9 publications

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

Dirichlet improvability for $S$-numbers

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