Continued fractions provide important methods to construct transcendental numbers. The first studies in this direction are due to Liouville, who dealt with unbounded partial quotients, then Maillet and Baker exhibited continued fractions with bounded partial quotients converging to transcendental numbers. Furthermore, Baker's results have been recently improved by several other authors dealing also with palindromic and automatic sequences for the partial quotients. These results are mainly based on the application of Roth's theorem and the Subspace theorem.
In this talk, we briefly review these classical results and then we focus on their translations in the framework of p-adic numbers.
We recall the construction of p-adic continued fractions as well as the p-adic versions of Roth's theorem and Subspace theorem. Finally, we prove the transcendence of some families of p-adic continued fractions, providing also a quantitative version of Ridout’s theorem (the p-adic analogue
of Roth’s theorem) and a study on the growth of denominators of convergents of algebraic numbers, establishing a p-adic version of a well-known result of Davenport and Roth.
