Limit theorems for sums of products of consecutive partial quotients of continued fractions

TL;DR

This paper investigates the asymptotic behavior of sums of products of consecutive partial quotients in continued fractions, establishing laws of large numbers and fractal dimensions related to their growth rates.

Contribution

It provides new probabilistic and multifractal results on the growth rate of partial quotient products in continued fractions, extending classical number theory insights.

Findings

Lebesgue measure of deviations tends to zero

Almost sure limit of normalized sums established

Hausdorff dimension of specific sets characterized

Abstract

Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the improvements to Dirichlet's theorem (1842). We establish both the weak and strong laws of large numbers for the partial sums $S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results: \medskip \begin{itemize} \item For any $\epsilon>0$, the Lebesgue measure of the set $$\left\{x\in(0, 1): \left|\frac{ S_n(x)}{n\log^2 n}-\frac1{2\log2}\right|\geq \epsilon\right\}$$tends to zero as $n$ to infinity. \item For Lebesgue almost all $x\in (0,1)$, $$\lim\limits_{n\rightarrow \infty} \frac{S_n(x)-\max\limits_{1\leq i \leq n}a_i(x)a_{i+1}(x)}{n\log^2n}=\frac{1}{2\log2}.$$ \item The Hausdorff dimension of the set $$E(\phi):=\left\{x\in(0,1):\lim\limits_{n\rightarrow \infty}\frac{S_n(x)}{\phi(n)}=1\right\}$$ is determined for a range of increasing functions $\phi: \mathbb N\to \mathbb R^+$. \end{itemize}