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

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.

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.

DOI : 10.5802/jtnb.872
Classification : 11J83, 11J13, 11K60
Mots clés : Diophantine approximation, systems of linear forms, Khintchine–Groshev theorem.
Hussain, Mumtaz 1 ; Yusupova, Tatiana 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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Hussain, Mumtaz; Yusupova, Tatiana. A note on the weighted Khintchine-Groshev Theorem. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 385-397. doi : 10.5802/jtnb.872. http://archive.numdam.org/articles/10.5802/jtnb.872/

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