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

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 {a1, a2, · · · , an}, where a1 < a2 < · · · < an are positive integers and have same remainder when divided by n. And let ai = n · qi + r, qi ∈ 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 and n is an even number, the generalized Kolakoski words are uniformly recurrent for the alphabet Σn 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.