Related papers: On a Problem of Steinhaus

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha,2\alpha,\ldots,N\alpha$, for any integer $N$ and real number $\alpha$. This statement…

We study the process of bootstrap percolation on n x n permutation matrices, inspired by the work of Shapiro and Stephens [5]. In this percolation model, cells mutate (from 0 to 1) if at least two of their cardinal neighbors contain a 1,…

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $\mathbb{Z}^d$ with parameter $p$, known also as percolation of words.\ In 1995, I.\ Benjamini and H.\ Kesten proved that, for $d \geq 10$…

This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all…

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the…

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here.…

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$…

A theorem of Glasner says that if $X$ is an infinite subset of the torus $\mathbb{T}$, then for any $\epsilon>0$, there exists an integer $n$ such that the dilation $nX=\{nx: x \in \mathbb{T} \}$ is $\epsilon$-dense (i.e, it intersects any…

A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that…

Let $S_{0,n}$ be an $n$-punctured sphere. For $n\geq 4$, we construct a sequence $(\mathcal{X}_i)_{i\in\mathbb{N}}$ of finite rigid sets in the pants graph $\mathcal{P}(S_{0,n})$ such that $\mathcal{X}_1 \subset \mathcal{X}_2 \subset…

A result of Rosenthal says that for every $q>1$ and $n \in \mathbb{N}$ there is $N \in \mathbb{N}$ such that every sequence of $N$ distinct positive numbers contains, after a suitable translation and possible multiplication by $-1$, a…

For any sequence $u$, the extremal function $Ex(u, j, n)$ is the maximum possible length of a $j$-sparse sequence with $n$ distinct letters that avoids $u$. We prove that if $u$ is an alternating sequence $a b a b \dots$ of length $s$, then…

Let a tribonacci sequence be a sequence of integers satisfying $a_k=a_{k-1}+a_{k-2}+a_{k-3}$ for all $k\ge 4$. For any positive integers $k$ and $n$, denote by $f_k(n)$ the number of tribonacci sequences with $a_1, a_2, a_3>0$ and with…

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every…

We show that a non-piercing family of connected planar sets with bounded independence number can be stabbed with a constant number of points. As a consequence, we answer a question of Axenovich, Kie{\ss}le and Sagdeev about the largest…

Given a sequence $s=(s_1,s_2,\ldots)$ of positive integers, the inversion sequences with respect to $s$, or $s$-inversion sequences, were introduced by Savage and Schuster in their study of lecture hall polytopes. A sequence…

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we…

According to the Erd\H{o}s discrepancy conjecture, for any infinite $\pm 1$ sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any $\pm 1$ sequence $(x_1,x_2,...)$ and a discrepancy…

The three distance theorem (also known as the three gap theorem or Steinhaus problem) states that, for any given real number $\alpha$ and integer $N$, there are at most three values for the distances between consecutive elements of the…

In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erd\"os that measures the minimal gap size $f(k)$ in the sequence of integers at least one of whose prime factors exceeds $k$. This…