Let g be a dimension function and let Formula: see text be an approximation function. The Generalised Baker–Schmidt Problem (1970) concerns the Formula: see text-dimensional Hausdorff measure (Formula: see text-measure) of the set of Formula: see text-approximable points on non-degenerate manifolds. The problem relates the ‘size’ of the set of Formula: see text- 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). The convergence case, however, represents a challenging open problem, and progress thus far has been effectuated in limited cases only. For instance, for the parabola, it has been established under some restrictions on the dimension function Formula: see text and for monotonic approximating functions. We prove some related new results for the parabola; in particular, we show that the monotonicity assumption on a multivariable approximating function cannot be removed. We go on to study Veronese curves in higher dimensions. Using Gelfond’s Lemma and some general irreducibility considerations for integer polynomials, in dimension three we are able to generalise a recent result of Pezzoni (2020) regarding the convergence theory of Formula: see text-measure.
Hussain et al. (Wed,) studied this question.