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.


11J25 Diophantine inequalities
Full Text: DOI


[1] Hartman, S; Szüsz, P, On congruence classes of denominators of convergents, Acta arith., 6, 179-184, (1960) · Zbl 0094.25804
[2] Szüsz, P, Über die metrische theorie der diophantischen approximation, II, Acta arith., 8, 225-241, (1963) · Zbl 0123.04602
[3] Duffin; Schaeffer, Khintchene’s problem in metric Diophantine approximation, Duke math. J., 8, 243-255, (1941) · Zbl 0025.11002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.