Limit theorems for sub-sums of partial quotients of continued fractions

Abstract

This paper studies the limit behaviour of sums of the form Tn(x)=∑1≤j≤nckj(x),(n=1,2,...)where (cj(x))j≥1 is the sequence of partial quotients in the regular continued fraction expansion of the real number x and (kj)j≥1 is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational α, the sequence (kjα)j≥1 is uniformly distributed modulo one and (kj)j≥1 is good universal. It was observed by the second author, for this class of sequences (kj)j≥1 that we have limn→∞Tn(x)n=+∞ almost everywhere with respect to Lebesgue measure. The case kj=j(j=1,2,…) is classical and due to A. Ya. Khinchin. Building on work of H. Diamond, Khinchin, W. Philipp, L. Heinrich, J. Vaaler and others, in the special case where kj=j(j=1,2,…,) we examine the asymptotic behaviour of the sequence (Tn(x))n≥1 in more detail.