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- W4288600716 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." @default.
- W4288600716 created "2022-07-29" @default.
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- W4288600716 date "2019-02-06" @default.
- W4288600716 modified "2023-09-24" @default.
- W4288600716 title "Diophantine approximation on curves" @default.
- W4288600716 doi "https://doi.org/10.48550/arxiv.1902.02094" @default.
- W4288600716 hasPublicationYear "2019" @default.
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