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- W1973482782 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." @default.
- W1973482782 created "2016-06-24" @default.
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- W1973482782 date "2006-11-14" @default.
- W1973482782 modified "2023-10-16" @default.
- W1973482782 title "A note on simultaneous Diophantine approximation on planar curves" @default.
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- W1973482782 doi "https://doi.org/10.1007/s00208-006-0055-1" @default.
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