A note on the weighted Khintchine-Groshev Theorem

Hussain, Mumtaz; Yusupova, Tatiana

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

Soit $W(m,n;\underline{\psi })$ l’ensemble des points ${\psi }_1,...,{\psi }_n$– approximables dans ${ℝ}^{mn}$. Le théorème classique de Khintchine–Groshev suppose une condition de monotonicité sur la fonction approximante $\underline{\psi }$. 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 $W(m,n;\underline{\psi })$ denote the set of ${\psi }_1,...,{\psi }_n$–approximable points in ${ℝ}^{mn}$. The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underline{\psi }$. 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.

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

[1] V. Beresnevich, V. Bernik, M. Dodson and S. Velani, Classical metric Diophantine approximation revisited in Analytic number theory, (2009) Cambridge Univ. Press, 38–61. | MR | Zbl

[2] V. Beresnevich, D. Dickinson and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc., 179(846), (2006), x+91. | MR | Zbl

[3] V. Beresnevich, A. Haynes and S. Velani, Multiplicative zero-one laws and metric number theory, Acta Arith., 160,2 , (2013), 101–114. | MR | Zbl

[4] V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math.(2), 164,3, (2006), 971–992. | MR | Zbl

[5] V. Beresnevich and S. Velani, Schmidt’s theorem, Hausdorff measures, and slicing, Int. Math. Res. Not., 24, (2006), Art. ID 48794. | MR | Zbl

[6] V. Beresnevich and S. Velani, A note on zero-one laws in metrical Diophantine approximation, Acta Arith., 133, 4, (2008), 363–374. | MR | Zbl

[7] V. Beresnevich and S. Velani, Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem, Int. Math. Res. Not. IMRN, 1, (2010), 69–86. | MR | Zbl

[8] D. Dickinson and M. Hussain, The metric theory of mixed type linear forms, Int. J. Number Theory, 9, 1, (2013), 77–90. | MR | Zbl

[9] M. M. Dodson, Geometric and probabilistic ideas in the metric theory of Diophantine approximations, Uspekhi Mat. Nauk, 48, 5 (293), (1993), 77–106. | MR | Zbl

[10] R. J. Duffin and A. C. Schaeffer, Khintchine’s problem in metric Diophantine approximation, Duke Math. J., 8, (1941), 243–255. | MR | Zbl

[11] A. V. Groshev, Un théoreme sur les systèmes des formes lineaires, Doklady Akad. Nauk SSSR., 19, (1938), 151–152. | Zbl

[12] G. Harman, Metric number theory, London Mathematical Society Monographs New Series, The Clarendon Press Oxford University Press, New York, 18, (1998). | MR | Zbl

[13] M. Hussain and S. Kristensen, Metrical results on systems of small linear forms, Int. J. of Number theory, 9, 3, (2013), 769–782. | MR

[14] M. Hussain and J. Levesley, The metrical theory of simultaneously small linear forms, Funct. Approx. Comment. Math., 48, 2, (2013), 167–181. | MR

[15] A. Khintchine, Zur metrischen theorie der diophantischen approximationen, Math. Zeitschr., 24, 1, (1926), 706–714. | MR

[16] A. Khintchine, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92, 1-2, (1924), 115–125. | MR

[17] L. Li, Zero-one laws in simultaneous and multiplicative diophantine approximation, Mathematika, 59, 2, (2013), 321–332. | MR

[18] V. G. Sprindžuk, Metric theory of Diophantine approximations, V. H. Winston & Sons, Washington, D.C. Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman, Scripta Series in Mathematics, (1979). | MR

[19] T. Yusupova, Problems in metric Diophantine approximations, D. Phil thesis. University of York, U. K., (2011).