The critical exponent of the Arshon words

For each $α$ > 2 there is an infinite binary word with critical exponent $α$

For each $α> 2$ there is a binary word with critical exponent $α$.

Critical exponents of infinite balanced words

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated theorem prover Walnut.

$α$-数、丢番图指数与Sturmian字的分解

$α$-numbers, diophantine exponent and factorisations of sturmian words

该研究引入Sturmian字的$α$-数和形式截距概念，通过Rauzy图构建形式体系，推导其一般分解公式，还计算了重复函数与丢番图指数，为Sturmian字分析提供新工具。

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.

具有少量不同回文的二元词的临界指数

Critical exponent of binary words with few distinct palindromes

本文研究了包含少量不同回文的无限二元词，根据临界指数对其进行分类，扩展了Fici和Zamboni的结果，并发现具有18和20个回文的词是特定态射不动点的morphic images。

具有因子复杂度2n+1的词和最小临界指数

Words with factor complexity $2n+1$ and minimal critical exponent

本文确认了Shallit和Shur关于因子复杂度为2n+1的词具有最小临界指数的猜想，通过计算机辅助证明。

Critical exponent of infinite balanced words via the Pell number system

In a recent paper of Rampersad et al., the authors conjectured that the smallest possible critical exponent of an infinite balanced word over a 5-letter alphabet is $3/2$. We prove this result, using a formulation of first-order logic, the Pell number system, and a machine computation based on finite-state automata.

A note on the maximum number of $k$-powers in a finite word

A \emph{power} is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer; the power is also called a {\em $k$-power} and $k$ is its {\em exponent}. We prove that for any $k \ge 2$, the maximum number of different non-empty $k$-power factors in a word of length $n$ is between $\frac{n}{k-1}-Θ(\sqrt{n})$ and $\frac{n-1}{k-1}$. We also show that the maximum number of different non-empty power factors of exponent at least 2 in a length-$n$ word is at most $n-1$. Both upper bounds generalize the recent upper bound of $n-1$ on the maximum numb…

Every nonnegative real number is an abelian critical exponent

The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number $θ$ there exists an infinite word having abelian critical exponent $θ$. We also extend this result to the $k$…

在指数为四的二元生成群中证明 5-Engel 恒等式

Proving the 5-Engel identity in the 2-generator group of exponent four

本文研究 $B(2,4)$ 中 5-Engel 字的四次幂表示，借助平凡字递归可枚举的归约方法，将其缩短为长度至多五的 26 个四次幂乘积。

Circular critical exponents for Thue-Morse factors

We prove various results about the largest exponent of a repetition in a factor of the Thue-Morse word, when that factor is considered as a circular word. Our results confirm and generalize previous results of Fitzpatrick and Aberkane & Currie.

改进丢番图指数的谱

Spectrum of the refined Diophantine exponent

该研究从组合与拓扑视角分析作者提出的改进丢番图指数，证明三元字母表上其谱为[1,∞]，具刘维尔数集类拓扑性质，并给出多类无限字的具体示例。

The spectrum of the exponents of repetition

For an infinite word $\mathbf{x}$, Bugeaud and Kim introduced a new complexity function $\text{rep}(\mathbf{x})$ which is called the exponent of repetition of $\mathbf{x}$. They showed $1\le \text{rep}(\mathbf{x}) \le \sqrt{10}-\frac{3}{2}$ for any Sturmian word $\mathbf{x}$. Ohnaka and Watanabe found a gap in the set of the exponents of repetition of Sturmian words. For an irrational number $θ\in(0,1)$, let \[ \mathscr{L}(θ):=\{\text{rep}(\mathbf{x}):\textrm{$\mathbf{x}$ is an Sturmian word of slope $θ$}\}.\] In this article, we look into $\mathscr{L}(θ)$. The minimum of $\mathscr{L}(θ)$ is d…

反平方与临界指数

Antisquares and Critical Exponents

本文研究了不包含任意大反平方的无限二进制词，刻画了最小反平方，并确定了包含特定数量反平方的二进制词语言的重复阈值及好词数量的增长率。

二元富词的重复阈值

The repetition threshold for binary rich words

本文证明二元富词的重复阈值为 $2+\sqrt{2}/2$，通过结构定理验证了 Baranwal 和 Shallit 的猜想，并解决了 Vesti 的开放问题。

Abelian Powers and Repetitions in Sturmian Words

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $α$. vAs an analogue of the critical exponent, we introduce the abelian critical …

Extending Dekking's construction of an infinite binary word avoiding abelian $4$-powers

We construct an infinite binary word with critical exponent 3 that avoids abelian 4-powers. Our method gives an algorithm to determine if certain types of morphic sequences avoid additive powers. We also show that there are $Ω(1.172^n)$ binary words of length $n$ that avoid abelian 4-powers, which improves on previous estimates.

On square factors and critical factors of $k$-bonacci words on infinite alphabet

For any integer $k>2$, the infinite $k$-bonacci word $W^{(k)}$, on the infinite alphabet is defined as the fixed point of the morphism $φ_k:\mathbb{N}\rightarrow \mathbb{N}^2 \cup \mathbb{N}$, where \begin{equation*} φ_k(ki+j) = \left\{ \begin{array}{ll} (ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2, (ki+j+1)& \text{if } j =k-1. \end{array} \right. \end{equation*} The finite $k$-bonacci word $W^{(k)}_n$ is then defined as the prefix of $W^{(k)}$ whose length is the $(n+k)$-th $k$-bonacci number. We obtain the structure of all square factors occurring in $W^{(k)}$. Moreover, we prove that the cri…

On balanced sequences and their critical exponent

We study aperiodic balanced sequences over finite alphabets. A sequence vv of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y'. We show that the language of v is eventually dendric and we focus on return words to its factors. We develop a method for computing the critical exponent and asymptotic critical exponent of balanced sequences, provided the associated Sturmian sequence u has a quadratic slope. The method is based on looking for the shortest return words to bispecial factors in v. We illustrate our method on several examples; in particula…

An inequality for the number of periods in a word

We prove an inequality for the number of periods in a word x in terms of the length of x and its initial critical exponent. Next, we characterize all periods of the length-n prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of n, and use this result to show that our inequality is tight for infinitely many words x. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where x is overlap-free or squarefree.

Abelian Repetitions in Sturmian Words

We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of elementary Number Theory. We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality holds for the Fibonacci infinite word. We further prove that the longest prefix of the Fibonacci infinite word that is an abelian repetition of pe…

Dyck 词、模式避免和自动序列

Dyck Words, Pattern Avoidance, and Automatic Sequences

研究二进制序列中 Dyck 词的性质，证明 7/3-幂自由词有界嵌套深度，并刻画 Thue-Morse 词中 Dyck 因子及其计数。

二进制自动序列的重复阈值

Repetition Threshold for Binary Automatic Sequences

本文研究了二进制自动序列的重复阈值，证明了存在临界指数不超过7/3的序列，并扩展至斐波那契和特拉宾卡自动序列。