  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.

Number Theory