Lucian Ilie, Jeffrey Shallit
Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number alpha such that there exists an infinite word over a k-letter alphabet that avoids beta-powers for all beta>alpha. We generalize this concept to include the lengths of the avoided words. We give some conjectures supported by numerical evidence and prove one of these conjectures.
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The repetition threshold is the smallest real number $α$ such that there exists an infinite word over a $k$-letter alphabet that avoids repetition of exponent strictly greater than $α$. This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree $3$. Additionally, we present bounds for the repetition thresholds of trees with bounded maximum degrees.
The limit of repetition thresholds of rich sequences
本文验证了关于回文富序列重复阈值趋于2的猜想，即当字母表大小趋于无穷时，$d$元回文富序列的重复阈值极限为2。
The Repetition Threshold for Rote Sequences
本文研究因子复杂度为2n的Rote词，利用Walnut定理证明器和从态射生成自动机的新技术，证明了该类无限二进制词的重复阈值为5/2。
The repetition threshold for words on $n$ letters, denoted $\mbox{RT}(n)$, is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for $n$ letters, denoted $\mbox{CRT}_{\mbox{W}}(n)$, $\mbox{CRT}_{\mbox{I}}(n)$, and $\mbox{CRT}_{\mbox{S}}(n)$, respectively. Currie and the present authors conjectured that $\mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n…
The repetition threshold for ternary rich words
本文证明了无16/7次幂三字母富字词的结构定理，并据此推导出三字母富字词的重复阈值为1+1/(3-μ)≈2.25876324，其中μ是多项式x^3-2x^2-1的唯一实根。
The asymptotic repetition threshold of sequences rich in palindromes
该研究聚焦富含回文的rich序列，证明d>1的递归d元rich序列类的渐近重复阈值为2且与字母表大小无关，补充了序列重复阈值相关理论。
A small morphism giving Abelian repetition threshold less than 2
本文构造了一个长度为13的循环态射，作用于8字母字母表，使Abelian重复阈值小于1.8，改进了先前需巨大字母表的构造。
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。
For rational $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where $x$ is nonempty, $x'\in\{x,x^\mathrm{R}\}$, and $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mathrm{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mathrm{URT}(3)=\tfrac{7}{4}$. Then we show that $\mathrm{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mathrm{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conjecture for $k\in\{4,8,12\}.$
A word is called $β$-free if it has no factors of exponent greater than or equal to $β$. The repetition threshold $\mathrm{RT}(k)$ is the infimum of the set of all $β$ such that there are arbitrarily long $k$-ary $β$-free words (or equivalently, there are $k$-ary $β$-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all $β$ such that - there are arbitrarily long $k$-ary $β$-free circular words is called…
The repetition threshold of episturmian sequences
本文研究 d 元 episturmian 序列的重复阈值，证明其由 d-bonacci 序列达到，值为 2+1/(t-1)，其中 t 为特定多项式的唯一正根。
For a rational number $r$ such that $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where the word $x$ is nonempty, the word $x'$ is in $\{x,x^R\}$, and we have $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mbox{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mbox{URT}(3)=\tfrac{7}{4}$. Then we show that $\mbox{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mbox{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conje…
Let $A$ be an $a$-letter alphabet. We consider fractional powers of $A$-strings: if $x$ is a $n$-letter string, $x^r$ is a prefix of $xxxx...$ having length $nr$. Let $l$ be a positive integer. Ilie, Ochem and Shallit defined $R(a,l)$ as the infimum of reals $r>1$ such that there exist a sequence of $A$-letters without factors (substrings) that are fractional powers $x^{r'}$ where $x$ has length at least $l$ and $r'\ge r$. We prove that $1+\frac{1}{la}\le R(a,l)\le 1+\frac{c}{la}$ for some constant $c$.
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。
We consider a variation on a classical avoidance problem from combinatorics on words that has been introduced by Mousavi and Shallit at DLT 2013. Let $\texttt{pexp}_i(w)$ be the supremum of the exponent over the products of $i$ factors of the word $w$. The repetition threshold $\texttt{RT}_i(k)$ is then the infimum of $\texttt{pexp}_i(w)$ over all words $w\inΣ^ω_k$. Mousavi and Shallit obtained that $\texttt{RT}_i(2)=2i$ and $\texttt{RT}_2(3)=\tfrac{13}4$. We show that $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac14$ if $i$ is even and $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac16$ if $i$ is odd and $i\ge3$.
Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting open problems. We address lower bounds on the repetition threshold of infinite rich words over 2 and 3-letter alphabets, and construct a candidate infinite rich word over the alphabet $Σ_2=\{0,1\}$ with a small critical exponent of $2+\sqrt{2}/2$. This represents the first progress on an open problem of Vesti from 2017.
The critical exponent $E(\mathbf u)$ of an infinite sequence $\mathbf u$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf u$. By the famous Dejean's theorem, $E(\mathbf u) \geq 1+\frac1{d-1}$ for every $d$-ary sequence $\mathbf u$. We define the asymptotic critical exponent $E^*(\mathbf u)$ as the upper limit of the maximal repetition of factors of length $n$. We show that for any $d>1$ there exists a $d$-ary sequence $\mathbf u$ having $E^*(\mathbf u)$ arbitrarily close to $1$. Then we focus on the class of $d$-ary balanced sequences. In this class, the values $…
Antisquares and Critical Exponents
本文研究了不包含任意大反平方的无限二进制词，刻画了最小反平方，并确定了包含特定数量反平方的二进制词语言的重复阈值及好词数量的增长率。
A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions …
In this note we answer a question of Hurlbert about pebbling in graphs of high girth. Specifically we show that for every g there is a Class 0 graph of girth at least g. The proof uses the so-called Erdos construction and employs a recent result proved by Czygrinow, Hurlbert, Kierstead and Trotter. We also use the Czygrinow et al. result to prove that Graham's pebbling product conjecture holds for dense graphs. Finally, we consider a generalization of Graham's conjecture to thresholds of graph sequences and find reasonably tight bounds on the pebbling threshold of the sequence of d-dimensional…
Rainbow Threshold Graphs
本文定义了$k$-彩虹阈值图这一阈值图的推广形式，证明其不满足一阶逻辑0-1律，且渐近几乎必然地，$(k+1)$-彩虹阈值图与$k$-彩虹阈值图不同构。
Mock threshold graphs are a simple generalization of threshold graphs that, like threshold graphs, are perfect graphs. Our main theorem is a characterization of mock threshold graphs by forbidden induced subgraphs. Other theorems characterize mock threshold graphs that are claw-free and that are line graphs. We also discuss relations with chordality and well-quasi-ordering as well as algorithmic aspects.
We present a framework which allows a uniform approach to the recently introduced concept of pseudo-repetitions on words in the morphic case. This framework is at the same time more general and simpler. We introduce the concept of a pseudo-solution and a pseudo-rank of an equation. In particular, this allows to prove that if a classical equation forces periodicity then it also forces pseudo-periodicity. Consequently, there is no need to investigate generalizations of important equations one by one.
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