Large Deviation Principle for arithmetic functions in continued fraction expansion

TL;DR

This paper establishes a large deviation principle for the arithmetic mean of continued fraction digits, providing a new probabilistic understanding of their atypical behavior despite classical limit theorems not applying.

Contribution

It introduces a large deviations rate function for continued fraction digits, a novel result in the probabilistic analysis of continued fractions.

Findings

Existence of a large deviations rate function for continued fraction digits

Quantitative estimates of probabilities for deviations from typical behavior

Insights into the distribution of continued fraction digits beyond classical theorems

Abstract

Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity.