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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)

Abstract

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The Littlewood conjecture states that $\liminf _{q\to \infty } q\cdot \|q\alpha \|\cdot \|q\beta \|=0$ 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 $ \liminf _{q\to \infty } q\cdot \log q\cdot \log \log q\cdot \|q\alpha \|\cdot \|q\beta \|>0$ is of full dimension.

MSC classification

Secondary: 11J13: Simultaneous homogeneous approximation, linear forms

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 31 - 55
Copyright
Copyright © University College London 2012

References

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