Transcendence of $p$-adic continued fractions and a quantitative $p$-adic Roth theorem

TL;DR

This paper advances the understanding of $p$-adic continued fractions by proving convergence to transcendental or quadratic irrationals without restrictions, and establishes a quantitative $p$-adic Roth theorem with growth estimates for algebraic number convergents.

Contribution

It improves transcendence results for $p$-adic continued fractions and provides a quantitative $p$-adic Roth theorem with growth analysis of convergents.

Findings

Palindromic and quasi-periodic $p$-adic continued fractions converge to transcendental or quadratic irrationals.

Established a $p$-adic version of Davenport-Roth theorem on growth of denominators.

Provided a quantitative form of Ridout's theorem in the $p$-adic setting.

Abstract

In this paper, we improve some transcendence results for $p$--adic continued fractions. In particular, we prove that palindromic and quasi--periodic $p$--adic continued fractions converge either to transcendental numbers or quadratic irrationals, removing any restriction on the $p$--adic norm of the partial quotients (or convergents) considered in other works. Moreover, we provide a quantitative version of Ridout's theorem (the $p$--adic analogue of Roth's theorem), and we study the growth of denominators of convergents of algebraic numbers, establishing a $p$--adic version of a well--known result of Davenport and Roth.