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

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.