Diophantine approximation on curves

TL;DR

This paper advances the understanding of Diophantine approximation on curves by establishing new results on the Hausdorff measure of approximable points, especially on Veronese curves and planar curves, addressing open problems in the convergence case.

Contribution

It proves new results on the Hausdorff measure for Diophantine approximation on Veronese and planar curves, generalizing previous work and exploring irreducibility and monotonicity conditions.

Findings

Established Hausdorff measure results on Veronese curves in any dimension.

Generalized recent results of Pezzoni for the case n=3.

Demonstrated the necessity of monotonicity in approximation functions for planar curves.

Abstract

Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $\Psi$-approximable points on nondegenerate manifolds. The problem relates the `size' of the set of $\Psi$-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any nondegenerate manifold. The convergence case, however, represents a 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 the $\HH^g$-measure on Veronese curves in any dimension $n$. As a consequence of one of our results, we generalize recent results of Pezzoni [Acta Arith. 193 (2020), no. 3, 269-281] regarding $n=3$. This improvement evolves from a deeper investigation on general irreducibility considerations applicable in arbitrary dimensions. We further investigate the $\HH^g$-measure for convergence on planar curves. We show that the monotonicity assumption on a multivariable approximating function cannot be removed for planar curves.