Badziahin, D. and Harrap, S. (2017) 'Cantor-winning sets and their applications.', Advances in mathematics., 318 . pp. 627-677.
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.
|Full text:||(AM) Accepted Manuscript|
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|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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