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.

Reçu le : 2012-12-17
Accepté le : 2013-05-13
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.872
Classification : 11J83,  11J13,  11K60
Mots clés: Diophantine approximation; systems of linear forms; Khintchine–Groshev theorem.
@article{JTNB_2014__26_2_385_0,
     author = {Mumtaz Hussain and Tatiana Yusupova},
     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},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.872},
     mrnumber = {3320485},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_385_0/}
}
Mumtaz Hussain; Tatiana Yusupova. 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. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_385_0/

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