A generalization of Sturmian sequences: Combinatorial structure and transcendence. (English) Zbl 0953.11007

Risley, Rebecca N.; Zamboni, Luca Q.

Acta Arith. 95, No. 2, 167-184 (2000).

Summary: We study dynamical properties of a class of uniformly recurrent sequences on a \(k\)-letter alphabet with complexity \(p(n)=(k-1)n+1.\) These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of the (binary) Sturmian sequences of Morse and Hedlund. We describe two combinatorial methods for constructing characteristic Arnoux-Rauzy sequences. The first, which is the central idea of the paper, is a simple combinatorial algorithm for generating the bispecial words. This description seems new even in the Sturmian case. The second is a ‘dual’ reformulation of an algorithm due to Arnoux and Rauzy; we use it to obtain a characterization of primitive morphic Arnoux-Rauzy sequences.
As an application of the first combinatorial description, we show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form \(V^{2+\epsilon }\) and in the Sturmian case arbitrarily large subwords of the form \(V^{\frac {5+\sqrt 5}{2}}.\)
Using a recent combinatorial interpretation of Ridout’s Theorem due to S. Ferenczi and C. Mauduit, we prove that an irrational number whose base \(b\)-digit expansion is an Arnoux-Rauzy sequence on \(k\geq 2\) letters, is transcendental. This result is an extension of a theorem of Ferenczi and Mauduit for \(k\in \{2,3\},\) and yields a class of transcendental numbers of complexity \(p(n)=(k-1)n+1\).

MSC:

11B85	Automata sequences
68R15	Combinatorics on words
11J91	Transcendence theory of other special functions

Keywords:

uniformly recurrent sequences; alphabet; Sturmian sequences; Arnoux-Rauzy sequences; combinatorial algorithm; bispecial words; primitive morphic Arnoux-Rauzy sequences; transcendental numbers