Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 205.34902
Autor: Erdös, Paul
Title: On the distribution of the convergents of almost all real numbers. (In English)
Source: J. Number Theory 2, 425-441 (1970).
Review: Let n1 < n2 < ... be an infinite sequence of integers. The necessary and sufficient condition that for almost all \alpha infinitely many ni should occur among the convergents of \alpha is that sumooi = 1 \phi(ni)/n2i = oo, where \phi is the Euler \phi-function. The necessity is obvious, but the proof of the sufficiency is complicated. In fact it is proved that if sumooi = 1 \phi (ni)/n2i = oo then for every \epsilon > 0 and almost all \alpha |\alpha -a/ni| < \epsilon /n2i, (a,ni) = 1 has infinitely many solutions.
Classif.: * 11J25 Diophantine inequalities
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