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

作者：Ma, Liangang; Nair, Radhakrishnan^*

来源：Indagationes Mathematicae, 2017, 28(5): 913-927.

DOI：10.1016/j.indag.2017.06.006

• Partial quotients
• Distribution functions
• Mixing

This paper studies the limit behaviour of sums of the form @@@ T-n(x) = Sigma(1 <= j <= n) c(kj)(x), (n = 1, 2,...) @@@ where (c(j) (x))(j >= 1) is the sequence of partial quotients in the regular continued fraction expansion of the real number x and (k(j))(j >= 1) is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational alpha, the sequence (k(j)alpha)(j >= 1) is uniformly distributed modulo one and (k(j))(j >= 1) is good universal. It was observed by the second author, for this class of sequences (k(j))(j >= 1) that we have lim(n ->infinity)T(n)(x)/n = +infinity almost everywhere with respect to Lebesgue measure. The case k(j) = 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 k(j) = j (j = 1, 2,...,) we examine the asymptotic behaviour of the sequence (T-n(x))(n >= 1) in more detail.

• 出版日期2017-10
• 单位滨州学院