Multidimensional statistics for finite orbits of generalised continued fractions

TL;DR

This paper establishes probabilistic laws like Large Deviation and Central Limit Theorem for digit frequencies in multidimensional continued fractions, showing finite orbits reflect the typical ergodic behavior of infinite trajectories.

Contribution

It introduces multidimensional statistical results for continued fractions, extending classical theories to higher dimensions and finite orbits.

Findings

Finite trajectories capture ergodic behavior of infinite trajectories.

Multidimensional results include Large Deviation and Central Limit Theorem.

Applicable to various algorithms like Brun's and Jacobi--Perron.

Abstract

We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and Central Limit Theorem, including multidimensional results for random vectors. These results apply to classical multidimensional continued fraction transformations including Brun's algorithm and the Jacobi--Perron algorithm, and more generally for maps satisfying mild contraction hypothesis on the inverse branches. We prove in particular that the finite trajectories capture the generic ergodic behaviour of infinite trajectories.