Badly approximable points on planar curves and winning

Abstract

For any i,j>0 with i+j=1, let Bad(i,j) denote the set of points (x,y)∈R2 such that max⁡{‖qx‖1/i,‖qy‖1/j}>c/q for some positive constant c=c(x,y) and all q∈N. We show that Bad(i,j)∩C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of Bad(i,j) is winning for two dimensional Schmidt games.