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



Ma, Liangang and Nair, Radhakrishnan
(2017) Limit theorems for sub-sums of partial quotients of continued fractions. Indagationes Mathematicae, 28 (5). pp. 913-927.

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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→∞[Formula presented]=+∞ 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.

Item Type: Article
Uncontrolled Keywords: Partial quotients, Distribution functions, Mixing
Depositing User: Symplectic Admin
Date Deposited: 10 Aug 2017 09:50
Last Modified: 19 Jan 2023 06:58
DOI: 10.1016/j.indag.2017.06.006
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3008894