A note on the weighted Khintchine-Groshev Theorem
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 385-397.

Let W(m,n;ψ ̲) denote the set of ψ 1 ,...,ψ n –approximable points in mn . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions 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;ψ ̲) l’ensemble des points ψ 1 ,...,ψ n – approximables dans mn . Le théorème classique de Khintchine–Groshev suppose une condition de monotonicité sur la fonction approximante 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.

DOI: 10.5802/jtnb.872
Classification: 11J83, 11J13, 11K60
Keywords: Diophantine approximation; systems of linear forms; Khintchine–Groshev theorem.
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},
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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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