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THE MIXED SCHMIDT CONJECTURE IN THE THEORY OF DIOPHANTINE APPROXIMATION

Part of:Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  10 June 2011

Dzmitry Badziahin,
Jason Levesley and
Sanju Velani
Dzmitry Badziahin
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, U.K. (email: db528@york.ac.uk)
Jason Levesley
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, U.K. (email: jl107@york.ac.uk)
Sanju Velani
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, U.K. (email: slv3@york.ac.uk)

Abstract

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Let 𝒟=(dn)∞n=1 be a sequence of integers with dn≥2, and let (i,j) be a pair of strictly positive numbers with i+j=1. We prove that the set of x∈ℝ for which there exists some constant c(x)≧0 such that \[ \max \!\big \{|q|_\mathcal {D}^{1/i}, \|qx\|^{1/j}\big \} > c(x)/ q \quad \mbox {for all } q \in \mathbb {N} \] is one-quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. This, in turn, establishes the natural analogue of Schmidt’s conjecture within the framework of the de Mathan–Teulié conjecture, also known as the “mixed Littlewood conjecture”.

MSC classification

Secondary:11K60: Diophantine approximation11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

Information

Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011, pp. 239 - 245
Copyright
Copyright © University College London 2011

References

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