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.