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A NOTE ON WEIGHTED BADLY APPROXIMABLE LINEAR FORMS

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  10 June 2016

STEPHEN HARRAP  and
NIKOLAY MOSHCHEVITIN
STEPHEN HARRAP
Affiliation:
Department of Mathematical Sciences, Science Laboratories, Durham University, South Rd, Durham, DH1 3LE, United Kingdom e-mail: s.g.harrap@durham.ac.uk
NIKOLAY MOSHCHEVITIN
Affiliation:
Department of Mathematics and Mechanics, Moscow State University, Leninskie Gory 1, GZ MGU, 119991 Moscow, Russia e-mail: moshchevitin@rambler.ru
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Abstract

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We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11J13: Simultaneous homogeneous approximation, linear forms 11K60: Diophantine approximation
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 349 - 357
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. An, J., Badziahin-Pollington-Velani's theorem and Schmidt's game, Bull. London Math. Soc. 45 (4) (2013), 721–733.CrossRefGoogle Scholar
2. An, J., 2 dimensional badly approximable vectors and Schmidt's game, Duke Math. J. 165 (2) (2016), 267–284.CrossRefGoogle Scholar
3. Badziahin, D., Pollington, A. and Velani, S., On a problem in simultaneous Diophantine approximations: Schmidt's conjecture, Ann. Math. 174 (3) (2011), 1837–1883.CrossRefGoogle Scholar
4. Bugeaud, Y., Harrap, S., Kristensen, S. and Velani, S., On shrinking targets for $\mathbb{Z}$ m actions on tori, Mathematika 56 (2) (2010), 193–202.CrossRefGoogle Scholar
5. Bugeaud, Y. and Laurent, M., Exponents of homogeneous and inhomogeneous Diophantine approximation, Moscow Math. J. 5 (2005), 747–766.CrossRefGoogle Scholar
6. Cassels, J. W. S., An introduction to Diophantine approximation, Cambridge Tracts in Math., vol. 45 (Cambridge University Press, Cambridge, 1957).Google Scholar
7. Davenport, H., A note on Diophantine approximation II, Mathematika 11 (1964), 50–58.CrossRefGoogle Scholar
8. Einsiedler, M. and Tseng, J., Badly approximable systems of affine forms, fractals, and Schmidt games, J. Reine Angew. Math. 2011 (660) (2011), 83–97.CrossRefGoogle Scholar
9. Harrap, S., Twisted inhomogeneous Diophantine approximation and badly approximable sets, Acta Arith. 151 (2012), 55–82.CrossRefGoogle Scholar
10. Khintchine, A. J., Einige Sátze šber Kettenbrúche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125.CrossRefGoogle Scholar
11. Kim, D. H., The shrinking target property of irrational rotations, Nonlinearity 20 (7) (2007), 1637–1643.CrossRefGoogle Scholar
12. Kleinbock, D., Badly approximable systems of affine forms, J. Number Theory 79 (1999), 83–102.CrossRefGoogle Scholar
13. Kleinbock, D. and Weiss, B., Modified Schmidt games and Diophantine approximation with weights, Adv. Math. 223 (2010), 1276–1298.CrossRefGoogle Scholar
14. Kronecker, L., Näherungsweise ganzzahlige Auflösung linearer Gleichungen (reprinted in Werke, B. G. Teubner, Leipzig, 1930).Google Scholar
15. Lagarias, J. C., Best Diophantine approximations to a set of linear forms, J. Austral. Math. Soc. Ser. A 34 (1983), 114–122.CrossRefGoogle Scholar
16. Moshcheivitin, N. G., A note on badly approximable affine forms and winning sets, Mosc. Math. J. 11 (1) (2011), 129–137.CrossRefGoogle Scholar
17. Moshchevitin, N. G., On Harrap's conjecture in Diophantine approximation. Preprint available at arXiv:1204.2561 (2012).Google Scholar
18. Pollington, A. and Velani, S., On simultaneously badly approximable pairs, J. London Math. Soc. 66 (2002), 29–40.CrossRefGoogle Scholar
19. Schmidt, W. M., On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 623 (1966), 178–199.CrossRefGoogle Scholar
20. Schmidt, W. M., Badly approximable systems of linear forms, J. Number Theory 1 (1969), 139–154.CrossRefGoogle Scholar
21. Schmidt, W. M., Open problems in Diophantine approximation, in Approximations diophantiennes et nombres transcendants (Luminy, 1982), Prog. Math., vol. 31 (Bertrand, D. and Waldschmidt, M., Editors) (Birkháuser, Basel, 1983), 271–287.Google Scholar
22. Tseng, J., Badly approximable affine forms and Schmidt games, J. Number Theory 129 (2009), 3020–3025.CrossRefGoogle Scholar
23. Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (3) (1916), 313–352.CrossRefGoogle Scholar