The complexity of smooth words on 2-letter alphabets

Abstract

Let γa,b(n) be the number of smooth words of length n over the alphabet {a,b} with a<b. Say that a smooth word w is left fully extendable (LFE) if both aw and bw are smooth. In this paper, we prove that for any positive number ξ and positive integer n0 such that the proportion of b’s is larger than ξ for each LFE word of length exceeding n0, there are two constants c1andc2 such that for each positive integer n, one has c1⋅nlog(2b−1)log(1+(a+b−2)(1−ξ))<γa,b(n)<c2⋅nlog(2b−1)log(1+(a+b−2)ξ). Moreover, if both a and b are even, then there are two suitable constants c1,c2 such that c1⋅nlog(2b−1)log((a+b)/2)<γa,b(n)<c2⋅nlog(2b−1)log((a+b)/2) for each positive integer n.