Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-19T09:11:28.458Z Has data issue: false hasContentIssue false
  • English
  • Français

ON MULTIPLICATIVELY BADLY APPROXIMABLE NUMBERS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  12 June 2012

Dzmitry A. Badziahin
Dzmitry A. Badziahin*
Affiliation:
Department of Mathematics, Durham University, Science Laboratories, South Rd., Durham, DH1 3LE, U.K. (email: dzmitry.badziahin@durham.ac.uk)
Get access

Abstract

The Littlewood conjecture states that for all (α,β)∈ℝ2. We show that with the additional factor of log q⋅log log q, the statement is false. Indeed, our main result implies that the set of (α,β) for which is of full dimension.

MSC classification

Secondary: 11J13: Simultaneous homogeneous approximation, linear forms
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 31 - 55
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Badziahin, D., Pollington, A. and Velani, S., On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture. Ann. of Math. (2) 174 (2011), 1837–1883.CrossRefGoogle Scholar
[2]Badziahin, D. and Velani, S., Multiplicatively badly approximable numbers and generalised Cantor sets. Adv. Math. 228(5) (2011), 2766–2796.CrossRefGoogle Scholar
[3]Bugeaud, Y. and Moshchevitin, N., Badly approximable numbers and Littlewood-type problems. Math. Proc. Cambridge Philos. Soc. 150 (2011), 215–226.CrossRefGoogle Scholar
[4]Einsiedler, M., Katok, A. and Lindenstrauss, E., Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164 (2006), 513–560.CrossRefGoogle Scholar
[5]Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons (Chichester, 1990).Google Scholar
[6]Pollington, A. and Velani, S., On a problem in simultaneously Diophantine approximation: Littlewood’s conjecture. Acta Math. 66 (2000), 29–40.Google Scholar