Badly approximable points on planar curves and a problem of Davenport

Let C be two times continuously differentiable curve in R 2 with at least one point at which the curvature is non-zero. For any i , j ⩾ 0 with i + j = 1 , let Bad ( i , j ) denote the set of points ( x , y ) ∈ R 2 for which max { ‖ q x ‖ 1 / i , ‖ q y ‖ 1 / j } > c / q for all q ∈ N . Here c = c ( x , y ) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.