The arithmetic of continued fractions in the field of $p$-adic numbers

TL;DR

This paper develops a comprehensive method for performing arithmetic with continued fractions over the field of p-adic numbers, revealing unique properties and measure-theoretic results distinct from real continued fractions.

Contribution

It introduces a complete methodology for p-adic continued fractions of transformations and analyzes their properties, including measure-zero exceptions.

Findings

Arithmetic operations on p-adic continued fractions are established.

Knowledge of partial quotients does not always suffice to recover transformations.

The set of elements with non-recoverable partial quotients has Haar measure zero.

Abstract

Continued fractions have been long studied due to their strong properties, such as rational approximation. In this extent, their arithmetic over real numbers has represented an intriguing problem throughout the years. In this paper, we develop the arithmetic of continued fractions over the field of $p$-adic numbers. In particular, we provide a complete methodology to compute the $p$-adic continued fraction of the M\"obius transformation and the bilinear fractional transformation of $p$-adic numbers. These allow any standard arithmetic operation over $p$-adic numbers to be performed. In great contrast with real continued fractions, we prove that the knowledge of arbitrarily many partial quotients of the initial continued fractions is not always sufficient to recover some partial quotients of the transformations. However, we prove that the set of elements for which this is not possible has Haar measure zero in $\mathbb{Q}_p$.