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

TL;DR

This paper generalizes the study of smooth infinite words over n-letter alphabets with specific modular properties, establishing new results on letter frequency, recurrence, and structural properties, extending prior work from 2-letter alphabets.

Contribution

It introduces new methods to analyze smooth infinite words over n-letter alphabets with the same remainder modulo n, generalizing previous 2-letter alphabet results.

Findings

Letter frequency of well-proportioned words is 1/n for r=0

Smooth infinite words are recurrent

Generalized Kolakoski words are uniformly recurrent under certain conditions

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.