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We prove that these Cantor sets are made up of transcendental numbers, apartfrom their endpoints $0$ and $1$, under some arithmetical assumptions on thedata. To that purpose, we establish a criterion of linear independence over thefield of algebraic numbers for the three numbers $1$, a characteristic Sturmiannumber, and an arbitrary Sturmian number with the same slope.

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