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Restivo Salemi property for $α$-power free languages with $α\geq 5$ and $k\geq 3$ letters

Josef Rukavicka

TL;DR

This work resolves a Restivo–Salemi-type conjecture for a broad class of power-free languages by proving the property for $L_{k,\alpha}$ with $\alpha \ge 5$ and $k \ge 3$. It leverages a construction of bi-infinite $\alpha$-power-free words containing a non-recurrent letter, building on prior results about extendability and the existence of appropriate left/right infinite words. The authors introduce a $\Gamma/\Delta$ framework and use König’s lemma to control factor occurrences, culminating in a main theorem that ensures any pair of finite factors can be embedded into a bi-infinite $L_{k,\alpha}$ word, and a corollary establishing the Restivo–Salemi property for these parameters. The results extend understanding of extendability in power-free languages and provide a constructive route to transition words, with implications for the theory of infinite words and combinatorics on words.

Abstract

In 2009, Shur published the following conjecture: Let $L$ be a power-free language and let $e(L)\subseteq L$ be the set of words of $L$ that can be extended to a bi-infinite word respecting the given power-freeness. If $u, v \in e(L)$ then $uwv \in e(L)$ for some word $w$. Let $L_{k,α}$ denote an $α$-power free language over an alphabet with $k$ letters, where $α$ is a positive rational number and $k$ is positive integer. We prove the conjecture for the languages $L_{k,α}$, where $α\geq 5$ and $k\geq 3$.

Restivo Salemi property for $α$-power free languages with $α\geq 5$ and $k\geq 3$ letters

TL;DR

This work resolves a Restivo–Salemi-type conjecture for a broad class of power-free languages by proving the property for with and . It leverages a construction of bi-infinite -power-free words containing a non-recurrent letter, building on prior results about extendability and the existence of appropriate left/right infinite words. The authors introduce a framework and use König’s lemma to control factor occurrences, culminating in a main theorem that ensures any pair of finite factors can be embedded into a bi-infinite word, and a corollary establishing the Restivo–Salemi property for these parameters. The results extend understanding of extendability in power-free languages and provide a constructive route to transition words, with implications for the theory of infinite words and combinatorics on words.

Abstract

In 2009, Shur published the following conjecture: Let be a power-free language and let be the set of words of that can be extended to a bi-infinite word respecting the given power-freeness. If then for some word . Let denote an -power free language over an alphabet with letters, where is a positive rational number and is positive integer. We prove the conjecture for the languages , where and .
Paper Structure (3 sections, 9 theorems, 9 equations, 1 table)

This paper contains 3 sections, 9 theorems, 9 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bi-infinite $\alpha$-power words

Key Result

Proposition 2.2

(reformulation of 10.1007/978-3-030-48516-0_22) If $(k,\alpha)\in\Upsilon$, $v\in L_{k,\alpha}^{\mathbb{N},L}$, $z\in\mathop{\mathrm{Suf}}\nolimits(v)$, $x\in F_r(v)\cap\Sigma_k$, $s\in L_{k,\alpha}^{\mathbb{N},L}$, and $x\not\in F(s)$ then there is a finite word $u\in\Sigma_k^*$ such that $z\in\mat

Theorems & Definitions (23)

  • Definition 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 13 more