2010/11/19 by Yun Bao Huang, Huang, Yun Bao
Computer Science · Mathematics · #Algorithms and Data Compression #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #cs.DM #cs.FL #math.CO #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1011.4438
25 pages
arxiv created 2010/11/19 · openalex publication_date 2010/11/19 · arxiv updated 2010/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.