On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations

Abstract

We prove that these Cantor sets are made up of transcendental numbers, up to their endpoints 0 and 1, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers 1, xi(1), xi(2), where xi(1) and xi(2) are two arbitrary Sturmian numbers with the same slope.