A note on the complexity of C∞-words

AbstractLet γ(n) be the number of C∞-words of length n. Say that a C∞-word w is left doubly extendable (LDE) if both 1w and 2w are C∞. We show that for any positive real number ϕ and positive integer N such that the proportion of 2’s is greater than 12−ϕ in each LDE word of length exceeding N, there are positive constants c1 and c2 such that c1nlog3log((3/2)+ϕ+(2/N))<γ(n)<c2nlog3log((3/2)−ϕ) for all positive integers n. With the best value known for ϕ, and large N, this gives c1n2.7087<γ(n)