A note on the complexity of C∞C∞-words

Let γ(n)γ(n) be the number of C∞C∞-words of length nn. Say that a C∞C∞-word ww is left doubly extendable   (LDE) if both 1w1w and 2w2w are C∞C∞. We show that for any positive real number ϕϕ and positive integer NN such that the proportion of 2’s is greater than 12−ϕ in each LDE word of length exceeding NN, there are positive constants c1c1 and c2c2 such that c1nlog3log((3/2)+ϕ+(2/N))<γ(n)