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.

For any real pair i, j ≥ 0 with i + j = 1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x := (x 1 , x 2 ) ∈ R 2 for which there exists a positive constant c(x) such that max qx 1 1/i , qx 2 1/j > c(x)/q ∀ q ∈ N.A new characterization of Bad(i, j) in terms of 'well-approximable' vectors in the area of 'twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad x (i, j), the 'twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen from Bad(i, j). The main results naturally generalise the i = j = 1/2 work of Kurzweil.

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In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarník type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves.

Abstract. We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.