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 $(α,β)$-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 $\times2, \times3$ problem.

40 pages; 08/05/15 improvements to introduction and various typos corrected. 10/09/15 conversion of notation in Theorems 11 & 12 to match Schmidt's original. Various typos and readability improvements elsewhere. A couple of Remarks added