On Circular Threshold Words and Other Stronger Versions of Dejean's conjecture

TL;DR

The paper develops a comprehensive framework for constructing and verifying threshold words (TW) and circular threshold words (CTWS) in relation to Dejean’s conjecture. It combines descent morphisms, $({\varepsilon},\mathbf E)$-bounded generalizations, and context-dependent substitutions to build infinite families of TWs with cyclic shifts and to realize exponentially growing TW trees. It also provides practical, polynomial-time verification schemes for key conditions, enabling computational proof of existence for certain $n\ge5$ and $|{\bf A}_n|$. The work proposes both Exponential Growth TW Sets (РРДГС) and near-unit-exponent long factors constructions, with careful attention to combinatorial properties of factors, roots, and periods. These methods have potential to advance both theoretical understanding and computational verification of Dejean-type results across broader alphabets.

Abstract

Let the root of the word $w$ be the smallest prefix $v$ of $w$ such that $w$ is a prefix of $vvv...$. $per(w)$ is the length of the root of $w$. For any $n\ge5$, an $n$-ary threshold word is a word $w$ such that for any factor (subword) $v$ of $w$ the condition $\frac{|v|}{per(v)}\le\frac{n}{n-1}$ holds. Dejean conjecture (completely proven in 2009) states for $n\ge5$ that exists infinitely many of $n$-ary TWs. This manuscript is based on the author's student works (diplomas of 2011 (bachelor's thesis) and 2013 (master's thesis) years) and presents an edited version (in Russian) of these works with some improvements. In a 2011 work proposed new methods of proving of the Dejean conjecture for some odd cases $n\ge5$, using computer verification in polynomial time (depending on $n$). Moreover, the constructed threshold words (TWs) are ciclic/ring TWs (any cyclic shift is a TW). In the 2013 work, the proof method (of 2011) was improved by reducing the verification conditions. A solution for some even cases $n\ge6$ is also proposed. A 2013 work also proposed a method to construct stronger TWs, using a TW tree with regular exponential growth. Namely, the TWs, where all long factors have an exponent close to 1.

Figures (1)

• Figure 1: Здесь '$\diamond$' - переменный символ из ${\bf A}_\textit{n}\setminus\{\textbf{a}\}$, $jump=pt_1{-}points_\textit{i}[(pt_1{-}s_\textit{i})/P_\textit{i}]$, $\textit{J}_1=prew[pt_1{-}jump]$, $\textit{J}_2=prew[pt_2]$.

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