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 textth 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.