Automatic sequences and transcendence

Abstract

The aim of this chapter is to investigate the connections between automatic sequences and transcendence in fields of positive characteristic. In the real case, it is well known that the expansion in a given integer basis of a rational number is ultimately periodic, which implies that its complexity function is bounded. More generally, the expansion of an algebraic irrational number in a given integer basis is supposed to be normal, whereas real numbers having as expansion a sequence with a low complexity function are conjectured to be either rational, or transcendental. In particular, real numbers having as binary expansion a Sturmian sequence, or a fixed point of a substitution over a two-letter alphabet which is either of constant length or primitive, are transcendental. The situation is much simpler in the case of formal power series with coefficients in a finite field. Indeed it is possible to characterize algebraicity in a simple way: a formal power series is algebraic if and only if the sequence of its coefficients is automatic; this criterion is known as Christol, Kamae, Mendès France, and Rauzy’s theorem. In a similar vein, the continued fraction expansion of an algebraic number is supposed to be unbounded if this number is neither quadratic nor irrational; here again much more is known in the case of formal power series with coefficients in a finite field. For instance, examples of algebraic series with unbounded partial quotients can be produced. For more details, see Sec. 3.3.

This chapter has been written by V. Berthé