## 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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