Diophantine approximation on curves

Abstract

Let g be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the g-dimensional Hausdorff measure (H g -measure) of the set of ψ-approximable points on non-degenerate manifolds. The problem relates the 'size' of the set of ψ-approximable points with the convergence or divergence of a certain series. There are two variants of this problem, concerning simultaneous and dual approximation. In the dual settings, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any non-degenerate manifold. The convergence case, however, represents the major challenging open problem and progress thus far has been effectuated in limited cases only. In this paper, we discuss and prove several results on H g -measure on non-degenerate planar curves. We prove that the monotonicity assumption on the multivariable approximating function cannot be removed for planar curves. This is in contrast to the problem on hypersurfaces, in dimensions greater than 2, where it has been recently proven to be unnecessary [arXiv:1803.02314]. For the single valued approximating functions, along with many other results which improve the existence understanding of the GBSP on planar curves, we further address the GBSP for the case of Veronese curves in any dimension n, and provide a generalisation of a recent result of Pezzoni (arXiv:1809.09742).

Jarník Type Theorems on Manifolds

“…Hence, the results of [21,23] are not applicable to one-dimensional manifolds (curves). However, results of similar nature for the special class of Veronese curves have been obtained in [22]. For general planar curves, the best result is due to Huang.…”