Hausdorff theory of dual approximation on planar curves

Abstract

Abstract Ten years ago, Beresnevich–Dickinson–Velani [Mem. Amer. Math. Soc. 179 (2006), no. 846] initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in ℝ 2 {\mathbb{R}^{2}} , which represents the first complete theory of its kind for a general class of manifolds.