Cantor-winning sets and their applications

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.

Decaying and non-decaying badly approximable numbers

“…In this terminology, the main result of [3] states that certain algebraic badly approximable points are decaying. Bugeaud also asks the following question: The latter question has been recently answered in dimension 1 by a preprint of D. Badziahin and S. Harrap [1,Theorem 15]. 2 In this paper, we give a different proof of Badziahin-Harrap's result, which is valid in higher dimensions as well.…”

“…2 In this paper, we give a different proof of Badziahin-Harrap's result, which is valid in higher dimensions as well. Our proof shows that the set of badly approximable points which are 1 The condition "badly approximable" has been omitted in the statement of [3,Problem 4.4], but the question obviously makes no sense without it. 2 In the current version of their paper they do not mention this connection; we informed them of it by private communication while writing this paper.…”

Metric Diophantine Approximation: Aspects of Recent Work

“…Remark 7.5. Building upon the one-dimensional, generalised Cantor sets framework formulated in [7], an abstract 'metric space' framework of higher dimensional generalised Cantor sets, branded as 'Cantor winning sets', has recently been introduced in [5]. Projecting this framework onto the specific one-dimensional construction of Cantor rich sets given above, the definition of Cantor-winning sets reads as follows.…”

“…The latter key condition implies that d q (I q ) is no more than 8R −ε provided that 8R −ε < 1. Most recently, David Simmons has shown that the notion of Cantor winning as defined in [5] is equivalent to the notion of potential winning as defined in [50].…”

Bad($\mathbf{w}$) is hyperplane absolute winning

“…In fact, the 'winning' property can be strengthened to 'absolute winning' on applying [Nes13, Appendix B], see also [ABV18,Remark 7]. For higher dimensions, the first named author [Ber15] proved that for every w ∈ W d the set f −1 (Bad (w)) is Cantor winning (see also [BH17,Theorem B]). This result was then improved by the third named author [Yan19] in the following manner.…”