On an extension of Philipp's Limit Theorem on continued fractions

Due to a study of multi-dimensional Kronecker sequences, we realise the importance of the growth rate and the distribution of multiple sums of partial quotients in continued fraction expansions. This leads us to consider an extension of Philipp’s Limit Theorem, or the strong law of large numbers on continued fractions, to multiple sums of partial quotients. In this paper, we extend the first half of Philipp’s Limit Theorem as follows. Let [Formula: see text] be natural numbers, and let [Formula: see text] be any positive function defined on [Formula: see text] such that [Formula: see text] is non-decreasing and that [Formula: see text] Then, for Lebesgue-almost all [Formula: see text] we have [Formula: see text] where [Formula: see text] denotes the [Formula: see text]th partial quotient of [Formula: see text] In addition, we study the complexity of exceptional sets against our result and show that they have full Hausdorff dimension whenever [Formula: see text] grows at a polynomial rate.