Erdős Covering Systems
A covering system of the integers is a finite collection of arithmetic progressions whose union is the set Z. The study of these objects was initiated in 1950 by Erdős , and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called “minimum modulus problem” was resolved in 2015 by Hough , who proved that in every covering system with distinct moduli, the minimum modulus is at most 1016.
In this mini-course we will give a gentle introduction to the theory of covering systems, and describe some recent progress. In particular, we will give a short proof of Hough’s theorem, and discuss the number of minimal covering systems, the density of the uncovered set. We will also describe a general method for attacking problems about the existence (or otherwise) of covering systems with certain properties.
P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, On the Erdős covering problem: the density of the uncovered set, arXiv:1811.03547
P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The Erdős–Selfridge problem with square-free moduli, arXiv:1901.11465
P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The structure and number of Erdős covering systems, arXiv:1904.04806
P. Erdős, On integers of the form 2 k + p and some related problems, Summa Brasil. Math., 2 (1950), 113–123.
M. Filaseta, K. Ford, S. Konyagin, C. Pomerance and G. Yu, Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc., 20 (2007), 495–517.
R. Hough, Solution of the minimum modulus problem for covering systems, Ann. Math., 181 (2015), 361–382.