Continued fractions in the field of p-adic numbers

TL;DR

This paper surveys the development of p-adic continued fractions, highlighting key results, recent advances, and open problems in the field of p-adic number theory and Diophantine approximation.

Contribution

It provides a comprehensive overview of the main results and recent research developments in p-adic continued fractions, an area gaining increasing interest.

Findings

Summary of foundational definitions and results

Recent progress in understanding p-adic continued fractions

Open problems and future research directions

Abstract

Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued fractions defined over the field of $p$--adic numbers $\mathbb{Q}_p$, which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems.