Cantor-winning sets and their applications

TL;DR

This paper introduces Cantor-winning sets, a new class of sets with properties similar to classical winning sets, and demonstrates their relevance in Diophantine approximation and related conjectures.

Contribution

The paper develops the theory of Cantor-winning sets, establishing their properties and showing their applicability to problems in Diophantine approximation.

Findings

Cantor-winning sets have maximal Hausdorff dimension.

They are invariant under countable intersections and bi-Lipschitz maps.

Application to open problems like the Mixed Littlewood conjecture.

Abstract

We introduce and develop a class of \textit{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 framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and the $\times2, \times3$ problem.