## On the distribution of the convergents of almost all real numbers.(English)Zbl 0205.34902

Let $$n_1 < n_2 < \ldots$$ be an infinite sequence of integers. The necessary and sufficient condition that for almost all $$\alpha$$ infinitely many $$n_i$$ should occur among the convergents of $$\alpha$$ is that $$\sum^\infty_{i=1} \varphi(n_i)/n^2_i = \infty$$, where $$\varphi$$ is the Euler $$\varphi$$-function. The necessity is obvious, but the proof of the sufficiency is complicated. In fact it is proved that if $$\sum^\infty_{i=1} \varphi (n_i)/n^2_i= \infty$$ then for every $$\epsilon >0$$ and almost all $$\alpha$$ $|\alpha -a/n_i| < \epsilon /n^2_i, \quad (a,n_i) = 1$ has infinitely many solutions.

### MSC:

 11J25 Diophantine inequalities
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### References:

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