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.

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Erdős 覆盖系统

覆盖系统是等差数列的有限集合，其并集是整数集。Erdős 于 1950 年发起了对这些物体的研究，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，他问是否存在具有不同模量的覆盖系统，其最小模量是任意大的。这个问题在 2015 年由 Hough 解决，他表明在任何此类系统中，最小模数最多为 10 16 。本笔记的目的是对 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有关覆盖系统的其他几个问题。我们希望这种我们称之为失真方法的技术在其他组合设置中会有更多的应用。