The Khintchine constants for generalized continued fractions

摘要

Let Tp(x)=1/xp (mod 1) for 0p0=0.241485…, then there exists an ergodic invariant measure of the form ρpdx. Let an=⌊(1/Tpn−1(x))p⌋, n⩾1, where ⌊t⌋ is the integer part of t. If p=1, then a1,a2,…,an are the partial quotients of the classical continued fraction of x. For a real number q, we consider averages of an:K(p,q,n,x)=(a1a2⋯an)1/nifq=0,((a1q+a2q+⋯+anq)/n)1/qifq≠0.We show that (i) for almost every x, Kp,q:=limn→∞K(p,q,n,x)<∞ if and only if q<1/p, (ii) limp→∞(logKp,q)/p=1 if q=0 where log denotes the natural logarithm, (iii) limp→∞logKp,q/logp=1/|q| if q<0. The limiting behavior of Kp,q is investigated as p↓p0 with computer simulations.