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 ${a_1,a_2,…,a_n}$, 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.

💡 Research Summary

The paper extends the theory of infinite smooth words, originally developed for binary alphabets, to arbitrary finite alphabets $\Sigma_n={a_1,\dots ,a_n}$ under the condition that all letters share the same remainder $r$ when divided by $n$. Formally each letter is written as $a_i=nq_i+r$ with $q_i\in\mathbb N$ and $r\in{0,\dots ,n-1}$. This “uniform remainder” hypothesis guarantees that the differentiation operator $D$, which maps a word to the sequence of lengths of consecutive identical blocks, interacts with the alphabet in a highly regular way.

Four main results are proved.

1. Letter‑frequency theorem (Theorem 1). When $r=0$ (all letters are multiples of $n$) any infinite word that is twice differentiable and “well‑proportioned” (the frequencies of the letters converge) has each letter appearing with asymptotic frequency $1/n$. The proof uses the fact that $D^2(w)$ produces a block‑length sequence whose average is exactly $n$, which can be modeled as a uniform Markov chain. As a corollary, the generalized Kolakoski sequences on even binary alphabets have frequency $1/2$, confirming the conjectured behaviour in the original binary case.

2. Recurrence theorem (Theorem 2). Every smooth infinite word is recurrent: every finite factor occurs infinitely often. The argument proceeds by induction on the number of differentiations, exploiting the unbounded growth of the block‑length sequence produced by repeated applications of $D$. Because $D$ can be applied indefinitely, any finite factor can be forced to appear in arbitrarily large windows of the original word.

3. Uniform recurrence theorem (Theorem 3). If either $r=0$ or $r>0$ with $n$ even, then the generalized Kolakoski word on $\Sigma_n$ equipped with the natural cyclic order $a_1\to a_2\to\cdots\to a_n\to a_1$ is uniformly recurrent. Uniform recurrence means that for every length $L$ there exists a constant $M(L)$ such that any factor of length at least $M(L)$ contains every possible factor of length $L$. The proof shows that under the cyclic order the block lengths generated by $D$ are bounded both below and above by expressions involving $n$ and $r$, which yields a uniform bound on the gaps between occurrences of any given factor.

4. Non‑closure under non‑trivial permutations (Theorem 4). For words that are three times differentiable, the set of all finite factors $F(w)$ is not invariant under any non‑identity permutation of the alphabet. The authors demonstrate that $D^3$ creates a block‑length pattern that is sensitive to the ordering of letters; applying a non‑trivial permutation changes this pattern and introduces new factors not present in $F(w)$. Consequently, $F(w)$ is not closed under any such permutation.

Methodologically, the paper relies on a careful analysis of the differentiation operator $D$ and its iterates. When $r=0$, $D^2$ yields a sequence whose entries are all multiples of $n$, allowing the use of ergodic arguments to obtain the $1/n$ frequency result. For recurrence and uniform recurrence, the authors exploit the fact that $D$ preserves the cyclic structure of the alphabet and that the block‑lengths remain within a finite interval determined by $n$ and $r$. The non‑closure result follows from a combinatorial examination of how $D^3$ encodes the ordering of letters.

These theorems subsume the earlier results of Brlek, Jamet, Paquin, and Stipulanti (2008), which dealt only with the binary case and parity constraints. By moving to an $n$‑letter setting and introducing the uniform remainder condition, the paper provides a unified framework that explains why the Kolakoski‑type sequences exhibit balanced frequencies, why they are recurrent, and under which circumstances they become uniformly recurrent.

The work opens several avenues for future research. One natural direction is to drop the uniform remainder hypothesis and study alphabets where the letters have different remainders modulo $n$; this would likely break the regularity of $D$ and require new techniques. Another possibility is to consider weighted versions of the differentiation operator, where block lengths are transformed by a function before the next iteration, leading to richer dynamical systems. Finally, quantitative measures such as subword complexity and entropy for the generalized Kolakoski sequences remain largely unexplored and could shed light on their randomness properties.

In summary, the paper delivers a comprehensive generalization of smooth infinite word theory to arbitrary finite alphabets with a common remainder, establishing precise frequency, recurrence, and symmetry properties, and thereby deepening our understanding of Kolakoski‑type sequences in a broader combinatorial context.