# Large deviations for denominators of continued fractions

## TL;DR

This paper establishes an optimal exponential upper bound on the probability that the denominator of the nth convergent in regular continued fractions deviates from its mean, linking it to the dimension spectrum of Lyapunov exponents.

## Contribution

It provides the first exponential bound on the deviation probability for continued fraction denominators, connecting it to Lyapunov exponents and the dimension spectrum.

## Key findings

- Exponential upper bound on deviation probability is optimal.
- Bound relates to the dimension spectrum of Lyapunov exponents.
- Results improve understanding of the statistical behavior of continued fraction denominators.

## Abstract

We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{n\pi^2}{12\log2}$. The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation.