On a continued fraction algorithm in finite extensions of $\Q_p$ and its metrical theory

TL;DR

This paper introduces a continued fraction algorithm for finite extensions of _p, proving finiteness properties for small degrees and analyzing its metrical behavior using ergodic theory and averages.

Contribution

It generalizes existing algorithms in _p to finite extensions and establishes finiteness results and metrical properties for these new algorithms.

Findings

Finiteness property proven for small degree extensions.

Metrical properties analyzed using subsequence ergodic theory.

Continued fraction algorithms extended to finite extensions of _p.

Abstract

We develop a continued fraction algorithm in finite extensions of $\Q_p$ generalising certain algorithms in $\Q_p$, and prove the finiteness property for certain small degree extensions. We also discuss the metrical properties of the associated continued fraction maps for our algorithms using subsequence ergodic theory and moving averages.