Related papers: Bounds for Erd\H{o}s covering systems in global fu…

Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not…

We prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the largest modulus is at least 108. We also prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the least…

Applying geometric methods of $2$-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite…

In this note we investigate the asymptotic behavior of the number of maximum modulus points, of an entire function, sitting in a disc of radius $r$. In 1964, Erd\Humlaut{o}s asked whether there exists a non-monomial function so that this…

We consider the problem of covering multiple submodular constraints. Given a finite ground set $N$, a cost function $c: N \rightarrow \mathbb{R}_+$, $r$ monotone submodular functions $f_1,f_2,\ldots,f_r$ over $N$ and requirements…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

Erd\H{o}s asked many mathematical questions. Some lead to exciting research, others turned out to be easily solved. In this article, we provide evidence that one of his questions, Erd\H{o}s problem \#278 , has no general answer. We do so by…

It is well known that in an exact covering system in $\mathbb{Z}$, the biggest modulus must be repeated. Very recently, Kim gave an analogous result for certain quadratic fields, and Kim also conjectured that it must hold in any algebraic…

We employ analytic number theoretic techniques, specifically character sums and Weil type estimates, to study the covering radius of the generalized Zetterberg codes over all finite fields. Although the even and odd field cases require…

We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete…

The covering radius problem is a question in coding theory concerned with finding the minimum radius $r$ such that, given a code that is a subset of an underlying metric space, balls of radius $r$ over its code words cover the entire metric…

P. Erd\H{o}s conjectured in 1962 that on the ring $\mathbb{Z}$, every set of $n$ congruence classes in $\mathbb{Z}$ that covers the first $2^n$ positive integers also covers the ring $\mathbb{Z}$. This conjecture was first confirmed in 1970…

As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because…

Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum of isomorphic simple $\mathbb{Z}_p H$-modules such…

A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2 \, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k}) \quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ is called…

Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erd\H{o}s-Falconer distance problem: If $|A|\gg…

F-theory admits 7-branes with exceptional gauge symmetries, which can be compactified to give phenomenological four-dimensional GUT models. Here we study general supersymmetric compactifications of eight-dimensional Yang-Mills theory. They…

If $\rho$ denotes a finite dimensional complex representation of $\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(\rho)$ of vector valued modular forms for $\rho$ is free and of finite rank over the ring $M$ of scalar…

Let $F_g$ be a closed orientable surface of genus $g$. A set $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if…

Let $\mathcal{M}_{g,\epsilon}$ be the $\epsilon$-thick part of the moduli space $\mathcal{M}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover $\mathcal{M}_{g,\epsilon}$ is…