Twisted inhomogeneous Diophantine approximation and badly approximable sets

Abstract: For any real pair i, j geq 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) in R^2 for which there exists a positive constant c(x) such that max {||qx_1||^-i, ||qx_2||^-j} > c(x)/q for all q in N. A new characterization of the set 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), thetwisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen from the set Bad(i, j). The main results naturally generalise the i=j=1/2 work of Kurzweil.