On the transcendence of real numbers with a regular expansion.
This paper is concerned with the long-standing problem: ``How regular or random is the \(b\)-ary expansion of an algebraic irrational number?'' where \(b\) is any integer greater than one. It is well known that the \(b\)-ary expansion of a rational number is ultimately periodic. It has been conjectured that the \(b\)-ary expansion of an irrational number is totally random, and several results show that if such expansion is ``too regular'', then the number is transcendental. Transcendence results have been obtained from a recent purely combinatorial condition formulated by Ferenczi and Mauduit. In particular, this condition has been used to obtain the transcendence of real numbers whose \(b\)-ary expansion is a Sturmian sequence. Here, the authors generalize this result by using the Ferenczi-Mauduit condition to prove the transcendence of real numbers whose \(b\)-ary expansion is the coding of an irrational rotation on the circle with respect to a partition in two intervals or arises from a non-periodic three-interval exchange transformation.
scientific article; zbMATH DE number 2012761