PreprintCantor-winning sets and their applicationsD. Badziahin, S. HarrapAbstractWe introduce and develop a class of Cantor-winning sets that share the same amenable properties as the classical winning sets associated to Schmidt's \((\alpha,\beta)\)-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 broad-reaching framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and approximation by multiplicative semigroups of integers. Keywords: winning sets, Schmidt winning game, absolutely winning game, Cantor set.
This paper is available as a pdf (280kB) file. It is also on the arXiv: arxiv.org/abs/1503.04738
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