Smooth infinite words over $n$-letter alphabets having same remainder when divided by $n$

Abstract

Brlek et al. (2008) studied smooth infinite words and established some results on letter frequency, recurrence, reversal and complementation for 2-letter alphabets having same parity. In this paper, we explore smooth infinite words over $n$-letter alphabet $\{a_1,a_2,...,a_n\}$, where $a_1<a_2<...<a_n$ are positive integers and have same remainder when divided by $n$. And let $a_i=n\cdot q_i+r,q_i\in N$ for $i=1,2,...,n$, where $r=0,1,2,...,n-1$. We use distinct methods to prove that (1) if $r=0$, the letters frequency of two times differentiable well-proportioned infinite words is $1/n$, which suggests that the letter frequency of the generalized Kolakoski sequences is $1/2$ for 2-letter even alphabets; (2) the smooth infinite words are recurrent; (3) if $r=0$ or $r>0\text{ and }n$ is an even number, the generalized Kolakoski words are uniformly recurrent for the alphabet ${\Sigma }_n$ with the cyclic order; (4) the factor set of three times differentiable infinite words is not closed under any nonidentical permutation. Brlek et al.'s results are only the special cases of our corresponding results.

25 pages