Adamczewski Boris & Yann Bugeaud

A short proof of the transcendence of the Thue-Morse continued fractions

Let $a$ and $b$ two disctinct positive integers and let $(u_n)_{n\geq0}$ be the Thue-Morse sequence on the alphabet $\{a,b\}$. In 1998, M. Queffélec proved that the real number $\xi:=[0;u_0,u_1,u_2,...]$ is transcendental. The aim of this Note is to provide a new and shorter proof of M. Queffélec's result.