Liminf Sets in Simultaneous Diophantine Approximation

Faustin Adiceam

doi:10.5802/jtnb.949

Faustin Adiceam ¹

¹ Department of Mathematics, University of York YO10 5DD Heslington, United Kingdom

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 461-483.

Abstract
Résumé

Let $𝒬$ be an infinite set of positive integers. Denote by $W_{τ, n}^{*} (𝒬)$ the set of $n$ –tuples of real numbers simultaneously $τ$ –well approximable by infinitely many rationals with denominators in $𝒬$ but by only finitely many rationals with denominators in the complement of $𝒬$ . The Hausdorff dimension of the liminf set $W_{τ, n}^{*} (𝒬)$ is determined when $n \geq 1$ and $τ > 2 + 1 / n$ . A $p$ –adic analogue of the problem is also studied.

Soit $𝒬$ un ensemble infini d’entiers naturels non nuls. Soit $W_{τ, n}^{*} (𝒬)$ l’ensemble des $n$ –uplets de réels simultanément approchables à l’ordre $τ$ par une infinité de rationnels dont les dénominateurs sont dans $𝒬$ mais seulement par un nombre fini de rationnels dont les dénominateurs sont dans le complémentaire de $𝒬$ . Nous déterminons la dimension de Hausdorff de l’ensemble liminf $W_{τ, n}^{*} (𝒬)$ lorsque $n \geq 1$ et $τ > 2 + 1 / n$ . Un analogue $p$ –adique du problème considéré est également étudié.

Received: 2014-04-14
Accepted: 2014-09-04
Published online: 2017-01-02

MR: 3509720 | Zbl: 1415.11102

DOI: 10.5802/jtnb.949

Classification: 11J83, 11K60
Keywords: Diophantine approximation, liminf sets, Hausdorff dimension.

Author's affiliations:

Faustin Adiceam ¹

¹ Department of Mathematics, University of York YO10 5DD Heslington, United Kingdom

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Faustin Adiceam. Liminf Sets in Simultaneous Diophantine Approximation. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 461-483. doi : 10.5802/jtnb.949. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.949/

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