Multiplicatively badly approximable numbers and generalised Cantor sets

Badziahin, Dzmitry; Velani, Sanju

doi:10.1016/j.aim.2011.06.041

Multiplicatively badly approximable numbers and generalised Cantor sets

Badziahin, Dzmitry; Velani, Sanju

Authors

Dzmitry Badziahin

Sanju Velani

Abstract

Let p be a prime number. The p -adic case of the Mixed Littlewood Conjecture states that View the MathML sourceliminfq→∞q⋅|q|p⋅‖qα‖=0 for all α∈Rα∈R. We show that with the additional factor of View the MathML sourcelogqloglogq the statement is false. Indeed, our main result implies that the set of α for which View the MathML sourceliminfq→∞q⋅logq⋅loglogq⋅|q|p⋅‖qα‖>0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.

Citation

Badziahin, D., & Velani, S. (2011). Multiplicatively badly approximable numbers and generalised Cantor sets. Advances in Mathematics, 228(5), 2766-2796. https://doi.org/10.1016/j.aim.2011.06.041

Journal Article Type	Article
Publication Date	Dec 1, 2011
Deposit Date	May 30, 2011
Publicly Available Date	May 6, 2014
Journal	Advances in Mathematics
Print ISSN	0001-8708
Publisher	Elsevier
Peer Reviewed	Peer Reviewed
Volume	228
Issue	5
Pages	2766-2796
DOI	https://doi.org/10.1016/j.aim.2011.06.041
Keywords	Diophantine approximation, Littlewood Conjecture.

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Copyright Statement
This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 228, 5, 2011, 10.1016/j.aim.2011.06.041.