Some results on asymmetric approximation of irrational numbers (Q1184821)

Let \(\xi\) be a real irrational number with simple continued fraction expansion \([a_ 0,a_ 1,a_ 2,\ldots]\), and let \(p_ n/q_ n\) be its \(n\)-th convergent. The authors prove two theorems on the approximation of \(\xi\) by consecutive convergents which improve results and to a considerable extent solve the problems posed by \textit{J. Tong} [Proc. Edinb. Math. Soc., II. Ser. 31, 197-204 (1988; Zbl 0645.10008)]. Theorem 1: For any \(n\) and \(\tau>a^{-1}_{n+1}\) the inequalities \[ - {1\over\sqrt{1+4\tau}\cdot q^ 2_ i}<\xi-{p_ i\over q_ i}<{\tau\over\sqrt{1+4\tau}\cdot q^ 2_ i} \] hold for at least one \(i\in\{n-1,n,n+1\}\).