Limit theorems related to beta-expansion and continued fraction expansion

TL;DR

This paper establishes a central limit theorem and a law of the iterated logarithm for the number of partial quotients in continued fraction expansions, linked to the first n digits of the beta-expansion for any beta > 1.

Contribution

It generalizes previous results by Faivre and Wu from beta=10 to all beta > 1, connecting beta-expansions with continued fractions through limit theorems.

Findings

Proves a central limit theorem for k_n(x)

Establishes a law of the iterated logarithm for k_n(x)

Extends known results to all beta > 1

Abstract

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n \in \mathbb{N}$). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence $\{k_n, n \geq 1\}$, which generalize the results of Faivre and Wu respectively from $\beta =10$ to any $\beta >1$.