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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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