On Symmetrical and Asymmetric Diophantine Approximation by Continued Fractions

AbstractLet (Pn/Qn)n ≥ 0 be the sequence of regular continued fraction convergents of the real irrational number x. Define the approximation constantsdn, n ≥ 0 by dn = dn(x) ≔ Qn + 1|Qnx − Pn|, n ≥ 0. In this paper the distribution for almost all x of the sequences (dn − 1, dn)n ≥ 1 and (dn − 1, dn, dn + 1)n ≥ 1 is discussed. Furthermore, two recent theorems by Jingcheng Tong concerning arithmetical properties of the dn′s are generalized