A note on the weighted 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

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 2, page 385-397
  • ISSN: 1246-7405

Abstract

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Let W ( m , n ; 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 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.

How to cite

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Hussain, Mumtaz, and Yusupova, Tatiana. "A note on the weighted Khintchine-Groshev Theorem." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 385-397. <http://eudml.org/doc/275762>.

@article{Hussain2014,
abstract = {Let $W(m, n; \underline\{\psi \})$ denote the set of $\psi _1,\ldots ,\psi _n$–approximable points in $\mathbb\{R\}^\{mn\}$. 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.},
affiliation = {School of Mathematical and Physical Sciences The University of Newcastle Callaghan, NSW 2308, Australia; Department of Mathematics University of York Heslington,York, YO105DD, UK},
author = {Hussain, Mumtaz, Yusupova, Tatiana},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Diophantine approximation; systems of linear forms; Khintchine–Groshev theorem; Khintchine-Groshev theorem},
language = {eng},
month = {10},
number = {2},
pages = {385-397},
publisher = {Société Arithmétique de Bordeaux},
title = {A note on the weighted Khintchine-Groshev Theorem},
url = {http://eudml.org/doc/275762},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Hussain, Mumtaz
AU - Yusupova, Tatiana
TI - A note on the weighted Khintchine-Groshev Theorem
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/10//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 2
SP - 385
EP - 397
AB - Let $W(m, n; \underline{\psi })$ denote the set of $\psi _1,\ldots ,\psi _n$–approximable points in $\mathbb{R}^{mn}$. 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.
LA - eng
KW - Diophantine approximation; systems of linear forms; Khintchine–Groshev theorem; Khintchine-Groshev theorem
UR - http://eudml.org/doc/275762
ER -

References

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  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. Zbl1236.11064MR2508636
  2. V. Beresnevich, D. Dickinson and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc., 179(846), (2006), x+91. Zbl1129.11031MR2184760
  3. V. Beresnevich, A. Haynes and S. Velani, Multiplicative zero-one laws and metric number theory, Acta Arith., 160,2 , (2013), 101–114. Zbl1292.11085MR3105329
  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. Zbl1148.11033MR2259250
  5. V. Beresnevich and S. Velani, Schmidt’s theorem, Hausdorff measures, and slicing, Int. Math. Res. Not., 24, (2006), Art. ID 48794. Zbl1111.11037MR2264714
  6. V. Beresnevich and S. Velani, A note on zero-one laws in metrical Diophantine approximation, Acta Arith., 133, 4, (2008), 363–374. Zbl1229.11102MR2457266
  7. V. Beresnevich and S. Velani, Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem, Int. Math. Res. Not. IMRN, 1, (2010), 69–86. Zbl1241.11086MR2576284
  8. D. Dickinson and M. Hussain, The metric theory of mixed type linear forms, Int. J. Number Theory, 9, 1, (2013), 77–90. Zbl1269.11067MR2997491
  9. M. M. Dodson, Geometric and probabilistic ideas in the metric theory of Diophantine approximations, Uspekhi Mat. Nauk, 48, 5 (293), (1993), 77–106. Zbl0831.11039MR1258756
  10. R. J. Duffin and A. C. Schaeffer, Khintchine’s problem in metric Diophantine approximation, Duke Math. J., 8, (1941), 243–255. Zbl0025.11002MR4859
  11. A. V. Groshev, Un théoreme sur les systèmes des formes lineaires, Doklady Akad. Nauk SSSR., 19, (1938), 151–152. Zbl0019.05104
  12. G. Harman, Metric number theory, London Mathematical Society Monographs New Series, The Clarendon Press Oxford University Press, New York, 18, (1998). Zbl1081.11057MR1672558
  13. M. Hussain and S. Kristensen, Metrical results on systems of small linear forms, Int. J. of Number theory, 9, 3, (2013), 769–782. Zbl1311.11074MR3043613
  14. M. Hussain and J. Levesley, The metrical theory of simultaneously small linear forms, Funct. Approx. Comment. Math., 48, 2, (2013), 167–181. Zbl1311.11075MR3100138
  15. A. Khintchine, Zur metrischen theorie der diophantischen approximationen, Math. Zeitschr., 24, 1, (1926), 706–714. Zbl52.0183.02MR1544787
  16. A. Khintchine, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92, 1-2, (1924), 115–125. MR1512207
  17. L. Li, Zero-one laws in simultaneous and multiplicative diophantine approximation, Mathematika, 59, 2, (2013), 321–332. Zbl06193343MR3081774
  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). MR548467
  19. T. Yusupova, Problems in metric Diophantine approximations, D. Phil thesis. University of York, U. K., (2011). 

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