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Simultaneous Diophantine approximation on polynomial curves. (English) Zbl 1279.11076
Let $$\psi: {\mathbb N} \rightarrow {\mathbb R}_+$$ be a decreasing function tending to zero. A vector $$x \in {\mathbb R}^n$$ is said to be simultaneously $$\psi$$-approximable if the inequality $| q x -p | < \psi(| q |)$ has infinitely many solutions $$q \in {\mathbb Z}$$, $$p \in {\mathbb Z}^n$$. The authors consider simultaneously $$\psi$$-approximable points on curves in $${\mathbb R}^n$$ parametrised by integer polynomials, i.e., curves of the form $\Gamma = \{(x, P_1(x), \dots, P_{n-1}(x)) \in {\mathbb R}^n : x \in {\mathbb R}\},$ where the $$P_j$$’s are polynomials with integer coefficients.
The authors prove a zero-infinity law for the Hausdorff measure of the set of $$\psi$$-approximable points on $$\Gamma$$, under the additional assumption that $$\psi$$ decreases rapidly in terms of the maximum degree $$d$$ of the polynomials $$P_j$$. The decay condition on $$\psi$$ implies that the approximating points must themselves lie on the curve, which is what makes the proof work. As a corollary, the Hausdorff dimension of the set of $$\psi$$-approximable points for $$\psi(r) = r^{-\tau}$$ is shown to be equal to $$2/(d(\tau + 1))$$, provided that $$\tau \geq \max\{d-1, 1\}$$.

##### MSC:
 11J83 Metric theory 11J13 Simultaneous homogeneous approximation, linear forms 11J54 Small fractional parts of polynomials and generalizations
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