Diophantine approximation by continued fractions. (English) Zbl 0739.11026

Let \(\xi\) be an irrational real number with simple continued fraction expansion \(\xi=[a_ 0,a_ 1,a_ 2,\dots]\). Let \(p_ n/q_ n\) be its \(n\)-th convergent, and define \(M_ n\) by \(\xi-p_ n/q_ n=(-1)^ n/(M_ nq^ 2_ n)\). The author continues his research on relations among consecutive values of \(M_ n\) [Math. Z. 184, 151-153 (1983; Zbl 0497.10024); Proc. Am. Math. Soc. 105, 535-539 (1989; Zbl 0663.10007) and some more papers]. The main result of the paper under review is that upper (resp. lower) bounds for \(M_{n-1}\) and \(M_ n\) imply a certain lower (resp. upper) bound for \(M_{n+1}\).


11J70 Continued fractions and generalizations
11J04 Homogeneous approximation to one number
11A55 Continued fractions