Erd\H{o}s covering systems

TL;DR

This paper discusses covering systems, a collection of arithmetic progressions covering all integers, and introduces a new technique called the distortion method, which simplifies and strengthens previous results, with potential broader applications.

Contribution

The paper presents a simpler, stronger variant of Hough's method, called the distortion method, for analyzing covering systems and related combinatorial problems.

Findings

Hough proved the minimum modulus in covering systems is at most 10^{16}.

The distortion method offers a new approach to studying covering systems.

Potential for the distortion method to be applied in other combinatorial contexts.

Abstract

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 Erd\H{o}s 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.