No abstract

In this article, we prove that for a completely multiplicative function f from N * to a field K such that the set {p | f (p) = 1K and p is prime} is finite, the asymptotic subword complexity of f is Θ(n t ), where t is the number of primes p that f (p) = 0K , 1K . This proves in particular that sequences like ((−1) v 2 (n)+v 3 (n) )n are not k-automatic for k ≥ 2.

Continued fraction expansions of automatic sequences have been extensively studied during the last decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponent, as one can find in the works of Allouche-Peyrière-Wen-Wen, Bugeaud, and the first author. The present paper is motivated by the converse problem: to study continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue-Morse and period-doubling sequences respectively, and prove that they are congruent to algebraic series in Z [[x]] modulo 4. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo 4 are 2-automatic. Our approach is to first guess the explicit formulas of certain subsequences of (Pn(x)) and (Qn(x)), where Pn(x)/Qn(x) is the canonical representation of the truncated continued fractions, then prove these formulas by an intricate induction involving eight subsequences while exploiting the relations between these subsequences.

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in [Formula: see text] modulo 4. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo 4 are 2-automatic.