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 |
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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 |