Badly approximable points on planar curves and a problem of Davenport
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چکیده
Let C be two times continuously differentiable curve in R 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 for which max{∥qx∥, ∥qy∥} > 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.
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تاریخ انتشار 2015