Cantor-winning sets and their applications

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 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.