A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain; Tatiana Yusupova

doi:10.5802/jtnb.872

Mumtaz Hussain ¹; Tatiana Yusupova ²

¹ School of Mathematical and Physical Sciences The University of Newcastle Callaghan, NSW 2308, Australia
² Department of Mathematics University of York Heslington,York, YO105DD, UK

Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 385-397.

Abstract
Résumé

Let $W (m, n; \underset{̲}{ψ})$ denote the set of $ψ_{1}, ..., ψ_{n}$ –approximable points in $ℝ^{m n}$ . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underset{̲}{ψ}$ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$ . It can not be removed for $m = n = 1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m = 2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem.

Soit $W (m, n; \underset{̲}{ψ})$ l’ensemble des points $ψ_{1}, ..., ψ_{n}$ – approximables dans $ℝ^{m n}$ . Le théorème classique de Khintchine–Groshev suppose une condition de monotonicité sur la fonction approximante $\underset{̲}{ψ}$ . Différents auteurs ont pu supprimer cette condition pour différents $m$ et $n$ . Mais elle ne peut pas être supprimée quand $m = n = 1$ , Duffin et Schaeffer ayant donné un contre-exemple. Nous traitons le seul cas restant $m = 2$ , et donc toutes les conditions non-nécessaires dans le théorème de Khintchine–Groshev sont maintenant enlevées.

MR

DOI: 10.5802/jtnb.872

Classification: 11J83, 11J13, 11K60
Keywords: Diophantine approximation; systems of linear forms; Khintchine–Groshev theorem.

Author's affiliations:

Mumtaz Hussain ¹; Tatiana Yusupova ²

¹ School of Mathematical and Physical Sciences The University of Newcastle Callaghan, NSW 2308, Australia
² Department of Mathematics University of York Heslington,York, YO105DD, UK

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     title = {A note on the weighted {Khintchine-Groshev} {Theorem}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {385--397},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Mumtaz Hussain; Tatiana Yusupova. A note on the weighted Khintchine-Groshev Theorem. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 385-397. doi : 10.5802/jtnb.872. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.872/

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