The mixed Schmidt conjecture in the theory of Diophantine approximation

Badziahin, D.; Levesley, J.; Velani, S.

doi:10.1112/s0025579311002075

The mixed Schmidt conjecture in the theory of Diophantine approximation

Badziahin, D.; Levesley, J.; Velani, S.

Authors

D. Badziahin

J. Levesley

S. Velani

Abstract

Let xs1D49F=(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 xxs2208xs211D 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”.

Citation

Badziahin, D., Levesley, J., & Velani, S. (2011). The mixed Schmidt conjecture in the theory of Diophantine approximation. Mathematika, 57(02), 239-245. https://doi.org/10.1112/s0025579311002075

Journal Article Type	Article
Publication Date	Jul 1, 2011
Deposit Date	May 30, 2011
Publicly Available Date	May 6, 2014
Journal	Mathematika
Print ISSN	0025-5793
Electronic ISSN	2041-7942
Publisher	Wiley
Peer Reviewed	Peer Reviewed
Volume	57
Issue	02
Pages	239-245
DOI	https://doi.org/10.1112/s0025579311002075

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Copyright Statement
Copyright © University College London 2011. This paper has been published in a revised form subsequent to editorial input by Cambridge University Press in 'Mathematika' (57: 02 (2011) 239-245) http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8326180.