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: