Hausdorff theory of dual approximation on planar curves

TL;DR

This paper completes the Hausdorff measure theory for dual approximation on planar curves, resolving the long-standing open problem of the convergence case for all such curves in .

Contribution

It establishes the convergence part of the Hausdorff measure theory for dual approximation on all planar curves, a major open problem in the field.

Findings

Complete Hausdorff measure theory for dual approximation on planar curves

First full convergence theory for non-degenerate manifolds in 

Resolved a decade-long open problem in metric number theory

Abstract

Ten years ago, Beresnevich-Dickinson-Velani 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 $\mathbb{R}^2$, which represents the first complete theory of its kind for a general class of manifolds.