Erdős covering systems


A covering system is a finite collection of arithmetic progressionswhose union is the set of integers. The study of these objects was initiated byErdős in 1950, and over the following decades he asked many questions aboutthem. Most famously, he asked whether there exist covering systems with distinctmoduli whose minimum modulus is arbitrarily large. This problem was resolvedin 2015 by Hough, who showed that in any such system the minimum modulus isat most 1016. The purpose of this note is to give a gentle exposition of a simpler and strongervariant of Hough’s method, which was recently used to answer several other questionsabout covering systems. We hope that this technique, which we call thedistortion method, will have many further applications in other combinatorial settings.

Publication Title

Acta Mathematica Hungarica