On the transcendence of real numbers with a regular expansion. (English) Zbl 1052.11052

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.


11J81 Transcendence (general theory)
11B83 Special sequences and polynomials
68R15 Combinatorics on words
Full Text: DOI


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