On symmetric and asymmetric diophantine approximation by continued fractions (Q1320500)

Let \(P_ n/Q_ n\), \(n\geq 1\), be the sequence of regular continued fraction convergents of the real irrational \(x\). Set \(d_ n= d_ n(x)= Q_{n+1}| Q_ n x-P_ n|\), \(n\geq 1\). The author derives the asymptotic behaviour of the vectors \((d_ n, d_{n+1})\) and \((d_ n, d_{n+1}, d_{n+2})\), which asymptotic result is utilized for discussing the validity of the inequalities \[ -tR(t)/ Q_ n Q_{n+1}\leq x-P_ n/Q_ n\leq R(t)/Q_ n Q_{n+1} \] with specific function \(R(t)\), \(t\geq 1\), and for \(1\leq n\leq N\) as \(N\to+\infty\). The discussion throughout the paper is very elegant, several known results follow from the new ones as corollaries, and the survey part of the paper is of value on its own.