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\(p\)-adic variant of the convergence Khintchine theorem for curves over \(\mathbb Z_p\). (English) Zbl 1069.11027

Let \(p\) be a prime, \(\mathbb Q_p\) the field of \(p\)-adic numbers with the Haar measure \(\mu\), \(\mathbb Z_p\) the ring of \(p\)-adic integers, and \(| \cdot| _p\) the \(p\)-adic valuation. Let \(\Psi:\mathbb R\rightarrow\mathbb R^+\) be a monotonically decreasing function such that \(\sum_{h=1}^\infty h^3\Psi(h)\) converges. Let \(f_i:\mathbb Z_p\rightarrow\mathbb Z_p\;(i=1,2,3)\) be normal functions in the sense of Mahler such that the Wronski determinant of their derivatives is non-zero almost everywhere in \(\mathbb Z_p\). This means that the torsion of the curve \((f_1(x),f_2(x),f_3(x))\) is non-zero almost everywhere in \(\mathbb Z_p\). It is proved, that the set of \(x\in\mathbb Z_p\) such that the inequality \[ | a_0+a_1f_1(x)+a_2f_2(x)+a_3f_3(x)| _p<\Psi(\max| a_i| ) \] holds for infinitely many vectors \((a_0,a_1,a_2,a_3)\in\mathbb Z^4\), has zero Haar measure.

MSC:

11J61 Approximation in non-Archimedean valuations
11J83 Metric theory
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References:

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