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The \(k\)-dimensional Duffin and Schaeffer conjecture. (English) Zbl 0714.11048

The following \(k\)-dimensional analogue of the Duffin and Schaeffer conjecture is proved: Let \(k>1\) and let \(\{\alpha_ n\}\) denote a sequence of real numbers with \(0\leq \alpha_ n<1/2\), and suppose that the series \(\sum^{\infty}_{n=1}(\alpha_ n\phi (n)/n)^ k\) diverges. Then the inequalities \(\max (| x_ 1n-a_ 1|,...,| x_ kn-a_ k|)<\alpha_ n,\quad (a_ i,n)=1\quad (i=1,...,k)\) have infinitely many solutions for almost all \(x\in {\mathbb R}^ k\).
Reviewer: V.Ennola

MSC:

11K60 Diophantine approximation in probabilistic number theory
11J83 Metric theory
11J25 Diophantine inequalities
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References:

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