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

TL;DR

This paper demonstrates that certain Cantor sets from contracted rotation dynamics consist mostly of transcendental numbers, with specific arithmetical conditions ensuring this property, and establishes criteria for linear independence involving Sturmian numbers.

Contribution

It introduces a criterion for linear independence over algebraic numbers involving Sturmian numbers and shows that the Cantor sets in question are primarily transcendental.

Findings

Most elements of the Cantor sets are transcendental numbers.

A criterion for linear independence over algebraic numbers involving Sturmian numbers is established.

Endpoints 0 and 1 are algebraic, but other elements are transcendental under certain conditions.

Abstract

We prove that these Cantor sets are made up of transcendental numbers, apart from 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 $1$, a characteristic Sturmian number, and an arbitrary Sturmian number with the same slope.