论文标题
ERDS覆盖系统
Erdős covering systems
论文作者
论文摘要
覆盖系统是算术进程的有限集合，其结合是整数。对这些物体的研究是由ErdőS于1950年启动的，在接下来的几十年中，他询问了许多有关它们的问题。最著名的是，他问是否存在具有不同模量的覆盖系统，其最小模量很大。霍夫（Hough）在2015年解决了这个问题，后者表明，在任何这样的系统中，最低模量最多为$ 10^{16} $。 本说明的目的是温和地阐述了霍夫方法的简单，更强的变体，该方法最近被用来回答有关覆盖系统的其他几个问题。我们希望这种称为失真方法的技术将在其他组合设置中具有许多进一步的应用程序。
A covering system is 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 $10^{16}$. 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 the distortion method, will have many further applications in other combinatorial settings.
