A note on simultaneous Diophantine approximation on planar curves

Abstract

Let $$\mathcal{S}_{n}(\psi_{1},\dots,\psi_{n})$$ denote the set of simultaneously $$(\psi_{1},\dots,\psi_{n})$$ - approximable points in $$\mathbb{R}^{n}$$ and $$\mathcal{S}^{*}_{n}(\psi)$$ denote the set of multiplicatively ψ-approximable points in $$\mathbb{R}^{n}$$ . Let $$\mathcal{M}$$ be a manifold in $$\mathbb{R}^{n}$$ . The aim is to develop a metric theory for the sets $$ \mathcal{M} \cap \mathcal{S}_{n}(\psi_1,\dots,\psi_n) $$ and $$\mathcal{M} \cap \mathcal{S}^{*}_{n}(\psi) $$ analogous to the classical theory in which $$\mathcal{M}$$ is simply $$\mathbb{R}^{n}$$ . In this note, we mainly restrict our attention to the case that $$\mathcal{M}$$ is a planar curve $$\mathcal{C}$$ . A complete Hausdorff dimension theory is established for the sets $$\mathcal{C} \cap \mathcal{S}_{2}(\psi_{1},\psi_{2}) $$ and $$\mathcal{C} \cap \mathcal{S}^{*}_{2}(\psi) $$ . A divergent Khintchine type result is obtained for $$\mathcal{C} \cap \mathcal{S}_{2}(\psi_1,\psi_2) $$ ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $$\mathcal{C}$$ of $$\mathcal{C} \cap \mathcal{S}_{2}(\psi_1,\psi_2) $$ is full. Furthermore, in the case that $$\mathcal{C}$$ is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for $$\mathcal{C} \cap \mathcal{S}_{2}(\psi_1,\psi_2) $$ naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for $$\mathcal{C} \cap \mathcal{S}^{*}_{2}(\psi)$$ constitute the first of their type.