A note on badly approximabe sets in projective space

TL;DR

This paper extends Diophantine approximation results in real projective space by establishing a Jarník-type theorem for badly approximable points and an analogue of Khintchine's theorem for intrinsic approximation.

Contribution

It proves that badly approximable points in projective space have full Hausdorff dimension and establishes a Khintchine-type convergence theorem for intrinsic approximation.

Findings

Badly approximable points have full Hausdorff dimension in certain compact sets.

An analogue of Khintchine's theorem for intrinsic approximation is established.

The results complement recent Khintchine-type theorems in projective spaces.

Abstract

Recently, Ghosh \& Haynes \cite{HG} proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarn\'{\i}k-type result also holds for `badly approximable' points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine's theorem for convergence relating to `intrinsic' approximation of points in these compact sets.