S-arithmetic inhomogeneous Diophantine approximation on manifolds

Datta, Shreyasi; Ghosh, Anish

doi:10.1016/j.aim.2022.108239

Cited by 4 publications

(3 citation statements)

References 27 publications

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“…Our first theorem shows the classical Khintchine-Groshev theorem with the new S-arithmetic setting based on Equation (1.3). The analogs related to Equation (1.5) can be deduced from [15,16,4], which answer to the more delicate question suggested by Baker and Sprindzuk, which is related to Diophantine approximation on manifolds. The original work for the real case was accomplished in [13].…”

Section: S-arithmeticmentioning

confidence: 91%

A quantitative Khintchine-Groshev theorem for S-arithmetic Diophantine approximation

Han¹

2022

Preprint

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In [18], Schmidt studied a quantitative type of Khintchine-Groshev theorem. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition [1].In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see [15]). Hence for the sake of completeness, we also deal with the convergence case of the Khintchine-Groshev theorem, based on this new generalization.

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Section: S-arithmeticmentioning

confidence: 91%

A quantitative Khintchine-Groshev theorem for S-arithmetic Diophantine approximation

Han¹

2022

Preprint

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“…The well known Sprindžuk's conjecture was proved in local fields; see [39,42,28]. Recent developments in ν-adic Diophantine approximation can be found in [42,53,6,21,5] and references therein. We refer the reader to the survey [24,48] and [1,24,28,27,34,4,47,56,26,25] for more recent results in Diophantine approximation in positive characteristic.…”

Section: Introductionmentioning

confidence: 99%

Singular vectors in affine subspaces and $Ψ$-Dirichlet numbers

Datta¹,

Xu²

2022

Preprint

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We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside them in both R and function field over finite fields. Another result of this paper shows that the only ψ-Dirichlet numbers in a function field over a finite field are rational functions, unlike ψ-Dirichlet numbers in R. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field.

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“…In [6], an inhomogeneous Khintchine type theorem was established for affine subspaces, complementing the earlier work [3] for nondegenerate manifolds, see also [27] for more inhomogeneous results on affine subspaces. Further, an S-arithmetic inhomogeneous Khintchine type theorem for nondegenerate manifolds was established by Datta and the second named author [12].…”

Section: Introductionmentioning

confidence: 99%