Cantor-winning sets and their applications

Badziahin, D.; Harrap, S.

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.

Citation

Badziahin, D., & Harrap, S. (2017). Cantor-winning sets and their applications. Advances in Mathematics, 318, 627-677. https://doi.org/10.1016/j.aim.2017.07.027

• Journal Article Type: Article
• Acceptance Date: Jul 31, 2017
• Online Publication Date: Aug 28, 2017
• Publication Date: Aug 28, 2017
• Deposit Date: Dec 30, 2015
• Publicly Available Date: Aug 28, 2019
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 318
• Pages: 627-677
• DOI: https://doi.org/10.1016/j.aim.2017.07.027
• Public URL: https://durham-repository.worktribe.com/output/1395766
• Related Public URLs: http://arxiv.org/abs/1503.04738