Dual Diophantine Approximation on Planar Curves: General Hausdorff theory

Abstract

The general Hausdorff theory of Dual Diophantine approximation on manifolds was initiated by the work of Beresnevich, Dickinson, and Velani, in which they established that the set of ψ -approximable points on a manifold has full measure when a certain sum diverges. A decade later, Hussain established the convergence counterpart to the above result in the case of the parabola. Not long after, Huang proved a convergence result for all planar curves with regards to the Hausdorff s-measure. In this thesis, we extend Huang's convergence result to the Hausdorff g-measure for all planar curves.