Heights and transcendence of $p$--adic continued fractions

TL;DR

This paper investigates the heights and transcendence of Browkin $p$--adic continued fractions, providing new bounds and proving the transcendence of specific families using $p$--adic Roth-like results.

Contribution

It introduces new insights into the Browkin $p$--adic continued fractions, including a $p$--adic Euclidean algorithm and transcendence proofs for certain families.

Findings

Established upper bounds for heights of periodic $p$--adic continued fractions.

Proved transcendence of two families of $p$--adic continued fractions.

Connected $p$--adic Euclidean algorithm with continued fraction properties.

Abstract

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin $p$--adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then, we focus on the heights of some $p$--adic numbers having a periodic $p$--adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with $p$--adic Roth-like results, in order to prove the transcendence of two families of $p$--adic continued fractions.