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Diophantine approximation on non-degenerate curves with non-monotonic error function. (English) Zbl 1180.11025
The paper under review deals with Diophantine approximation in the linear forms setting of points on curves in Euclidean space. Let \(f_1, \dots, f_n: I \rightarrow {\mathbb R}\) be \(C^n\) functions where \(I \subseteq {\mathbb R}\) is some interval, and suppose that the Wronskian of the family is non-vanishing almost everywhere. Let \({\mathcal F}_n\) denote the set of functions \(a_n f_n(x) + \dots + a_1 f_1(x) + a_0\) where \((a_0, \dots, a_n) \in {\mathbb Z}^{n+1}\), let \(\Psi : {\mathbb R} \rightarrow {\mathbb R}_+\) be some function and let \(L_n(\Psi)\) be the set of \(x \in I\) for which \(| F(x) | < \Psi(H(F))\) for infinitely many \(F \in {\mathcal F}_n\), where \(H(F) = \max\{| a_1|, \dots, | a_n |\}\). In other words, the set \(L_n(\Psi)\) is the set of dually \(\Psi\)-approximable points on the curve \(\{(f_n(x), \dots, f_1(x)) \in {\mathbb R}^n : x \in I\}\).
If \(\Psi\) is monotonic the set \(L_n(\Psi)\) is null or full according to the convergence or divergence of the series \(\sum_{h=1}^\infty h^{n-1} \Psi(h)\). The divergence case of this statement is not valid without the monotonicity assumption on \(\Psi\). However, the authors show that assumption is not needed for the convergence case. This extends earlier work by V. Beresnevich [Acta Arith. 117, No. 1, 71–80 (2005; Zbl 1201.11078)], who proved this in the special case when \(f_i(x) = x^i\).
To prove their result, the authors split the set \(L_n(\Psi)\) up into four different sets, depending on the magnitude of the derivative of the approximating functions \(F \in {\mathcal F}_n\). Since each of these sets can be shown by different methods to be null under the convergence assumption, it is concluded that the set \(L_n(\Psi)\) is also null.

MSC:
11J83 Metric theory
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