Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778838 | Indagationes Mathematicae | 2017 | 15 Pages |
This paper studies the limit behaviour of sums of the form Tn(x)=â1â¤jâ¤nckj(x),(n=1,2,...)where (cj(x))jâ¥1 is the sequence of partial quotients in the regular continued fraction expansion of the real number x and (kj)jâ¥1 is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational α, the sequence (kjα)jâ¥1 is uniformly distributed modulo one and (kj)jâ¥1 is good universal. It was observed by the second author, for this class of sequences (kj)jâ¥1 that we have limnââTn(x)n=+â almost everywhere with respect to Lebesgue measure. The case kj=j(j=1,2,â¦) 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,â¦,) we examine the asymptotic behaviour of the sequence (Tn(x))nâ¥1 in more detail.