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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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