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Cantor-winning sets and their applications.

Badziahin, D. and Harrap, S. (2017) 'Cantor-winning sets and their applications.', Advances in mathematics., 318 . pp. 627-677.

Abstract

We introduce and develop a class of Cantor-winning sets that share the same amenable properties as the classical winning sets associated to Schmidt’s (α, β)-game: these include maximal Hausdorff dimension, invariance under countable intersections with other Cantor-winning sets and invariance under bi-Lipschitz homeomorphisms. It is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of Diophantine approximation fit nicely into our framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and the ×2,×3 problem.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.aim.2017.07.027
Publisher statement:© 2018 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Date accepted:31 July 2017
Date deposited:01 May 2018
Date of first online publication:28 August 2017
Date first made open access:28 August 2018

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