Erdős covering systems
Erdős covering systems
复制标题
ErdÅs 覆盖系统
DOI：
10.1007/s10474-020-01048-z
复制
发表时间：
2020
影响因子：
0.9
通讯作者：
Tiba, M.
中科院分区：
文献类型：
--
作者：
Balister, P.;Bollobás, B.;Morris, R.;Sahasrabudhe, J.;Tiba, M.
覆盖系统是算术级数的有限集合，其并集是整数的集合。对这些天体的研究是由Erdens于1950年发起的，在接下来的几十年里，他提出了许多关于它们的问题。最著名的是，他问是否存在覆盖系统与不同的模，其最小模是任意大的。这个问题在2015年由Hough解决，他证明了在任何这样的系统中，最小模至多为1016。本文的目的是对Hough方法的一个更简单、更强大的变体进行温和的阐述，该方法最近被用来回答关于覆盖系统的其他几个问题。我们希望，这种技术，我们称之为失真方法，将有许多进一步的应用在其他组合设置。
Acovering systemis a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is at most 1016.The purpose of this note is to give a gentle exposition of a simpler and stronger variant of Hough’s method, which was recently used to answer several other questions about covering systems. We hope that this technique, which we call thedistortion method, will have many further applications in other combinatorial settings.