Related papers: Asymptotic repetitive threshold of balanced sequen…

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have…

Let D be a Dyck path chosen uniformly from the set of Dyck paths with 2n steps. We prove that the sequence of the exponential moments of the maximum of D normalized by the square root of n converges in the limit of infinite n, and therefore…

The identification of binary sequences with large merit factor (small mean-squared aperiodic autocorrelation) is an old problem of complex analysis and combinatorial optimization, with practical importance in digital communications…

Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence…

Let $D=(V,E)$ be a strongly connected digraph and let $u ,v\in V(D)$. The maximum distance $md (u,v)$ is defined as\\ $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length of a…

Given a positive integer $n$, consider a random permutation $\tau$ of the set $\{1,2,\ldots, n\}$. In $\tau$, we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each…

An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has…

Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\em $\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\leq \alpha |u|$. Our main result is the $O(\alpha n)$ bound on the…

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…

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive discriminant. In this paper, we consider the Diophantine equation $u_m + u_n = a_1 n_1! + \cdots + a_k n_k!$ and prove that there are only finitely many…

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…

The function $\inf_n nx^{1/n}$ has the asymptotics $eu+e d^2(u)/(2u)+O(1/u^2)$ as $x\to\infty$, where $u=\log x$ and $d(u)$ is the distance from $u$ to the nearest integer. We generalize this observation. First, the curves $y=nx^{1/n}$ can…

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) $v$…

We prove a Berry-Esseen bound in de Jong's classical CLT for normalized, completely degenerate $U$-statistics, which says that the convergence of the fourth moment sequence to three and a Lindeberg-Feller type negligibility condition are…

A word~$w$ has a border $u$ if $u$ is a non-empty proper prefix and suffix of $u$. A word~$w$ is said to be \emph{closed} if $w$ is of length at most $1$ or if $w$ has a border that occurs exactly twice in $w$. A word~$w$ is said to be…

We study asymptotic anytime-valid confidence sequences for degree-two U-statistics under continuous monitoring. In the nondegenerate case, Hoeffding's projection reduces the problem to a time-uniform central limit theory for the partial…

The first two authors have shown [KK99,KK00] that the sum the exponent (and thus the number) of maximal repetitions of exponent at least 2 (also called runs) is linear in the length of the word. The exponent 2 in the definition of a run may…

Let $\{u_n\}_n$ be a non-degenerate linear recurrence sequence of integers with Binet's formula given by $u_n= \sum_{i=1}^{m} P_i(n)\alpha_i^n.$ Assume $\max_i \vert \alpha_i \vert >1$. In 1977, Loxton and Van der Poorten conjectured that…

We consider Rote words, which are infinite binary words with factor complexity $2n$. We prove that the repetition threshold for this class is $5/2$. Our technique is purely computational, using the Walnut theorem prover and a new technique…

We study how much injective morphisms can increase the repetitiveness of a given word. This question has a few possible variations depending on the meaning of ``repetitiveness''. We concentrate on fractional exponents of finite words and…