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A Khintchine-type version of Schmidt’s theorem for planar curves. (English) Zbl 0978.11033

The following theorem is proved: Let \(I\subset\mathbb R\) be an interval and suppose that the functions \(f_1,f_2:I\rightarrow\mathbb R\) are \(C^3\) and satisfy \(f_1'(x)f_2''(x)-f_1''(x) f_2'(x)\neq 0\) for almost all \(x\in I\). Let \(\psi:\mathbb N\rightarrow\mathbb R^+\) be a decreasing function and suppose that the sum \(\sum_{r=1}^\infty\psi(r)\) converges. Then, for almost all \(x\in I\), the inequality \[ |q_1f_1(x)+q_2f_2(x)+q_0|<\psi(H(\mathbf q))H(\mathbf q)^{-1} \] holds for only finitely many integer vectors \(\mathbf q=(q_0,q_1,q_2)\). Here \(H(\mathbf q)= \max\{|q_i|\}\). In a later paper [ibid. 455, 3053-3063 (1999; Zbl 0978.11031)], the authors together with V. V. Beresnevich have completed the theorem by showing that the result is true for infinitely many \(\mathbf q\) if the series diverges.

MSC:

11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory

Citations:

Zbl 0978.11031
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