Limit theorems on counting large continued fraction digits

Abstract: We establish a central limit theorem for counting large continued fraction digits $(a_n)$, that is, we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp, which additionally assumes that bn tends to infinity. Moreover, we also show this kind of central limit theorem for counting the number of occurrences entries such that the continued fraction entry lies between $d_n$ and $d_n(1+1/c_n)$ for given sequences $(c_n)$ and $(d_n)$. For such intervals we also give a refinement of the famous Borel–Bernstein theorem regarding the event that the nth continued fraction digit lying infinitely often in this interval. As a side result, we explicitly determine the first $\phi$-mixing coefficient for the Gauss system - a result we actually need to improve Philipp's theorem. This is joint work with Marc Kesseböhmer.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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