On the approximation by continued fractions. II
With the help of a powerful method developed by \textit{H. Jager} and the author [Indag. Math. 51, 289-307 (1989; Zbl 0695.10029)] which is based on the dynamical system underlying the continued fraction algorithm, the author gives short proofs of already known results on asymmetric diophantine approximation and he also gives answers to some open questions on this topic as for example the following solution of a problem settled by Tong. Theorem: Let \(\tau\geq 1\) and let x be a real irrational number with continued fraction expansion \(x=[a_ 0,a_ 1,a_ 2,...]\). Then from the three best approximation convergents \(p_ i/q_ i\), \(i=n,n+1,n+2\), \(n\geq 1\), at least one satisfies \[ -\frac{\tau}{\sqrt{a^ 2_{n+2}+4\tau}}\cdot \frac{1}{q^ 2_ i}<x-\frac{p_ i}{q_ i}<\frac{1}{\sqrt{a^ 2_{n+2}+4\tau}}\cdot \frac{1}{q^ 2_ i}. \] In case \(0<\tau <1\) there exists an irrational number x and a positive integer n such that \(p_ i/q_ i\) does not satisfy the inequality for \(i=2n,2n+1,2n+2\).
scientific article; zbMATH DE number 4150290
scientific article