A note on three problems in metric Diophantine approximation

TL;DR

This paper advances the Hausdorff measure theory in inhomogeneous metric Diophantine approximation by establishing new theorems for multiplicatively approximable points in the plane and on curves, and analyzing badly approximable points.

Contribution

It provides the first Hausdorff measure results for multiplicatively approximable points in the plane and on curves, extending classical Diophantine approximation theory.

Findings

Hausdorff measure theorem for multiplicatively approximable points in the plane

Hausdorff measure theorem for points on non-degenerate planar curves

Full Hausdorff dimension of inhomogeneously badly approximable points in the plane

Abstract

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider three inhomogeneous problems that further develop these classical results. Firstly, we obtain a Jarnik type theorem for the set of multiplicatively approximable points in the plane. This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting. Next, we establish a Jarnik type theorem for the set of multiplicatively approximable points on a non-degenerate planar curve. This completes the Hausdorff theory for planar curves. Finally, we show that the set of simultaneously inhomogeneously (i,j)-badly approximable points in the plane is of full dimension. The underlying philosophy behind the proof has other applications; e.g. towards establishing the inhomogeneous version of Schmidt's Conjecture. The higher dimensional analogues of the planar results are also discussed.