A note on badly approximabe sets in projective space (Q509992)

The first results regarding Diophantine approximation in projective space took place in the late 60's. The authors of this note provide a projective analogue to Jarník's theorem and continue the process of confirming that the theory of Diophantine approximation in projective space fits with the classical results. More precisely, they prove that the natural analogue in projective space of the classical set of badly approximable numbers when intersected with certain compact subsets, \(\Omega\), of the standard \(N\)-dimensional projective space over \(\mathbb{R}^{N+1}\), has full Hausdorff dimension inside \(\Omega\). It is also worth mentioning that the proof uses an analogue of the ``symplex lemma'' from classical Diophantine approximation. Finally, an analogue of the ``convergence'' part of Khintchine's theorem for approximation inside these compact sets \(\Omega\) is given.