On Erdős covering systems in global function fields

Abstract

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most $10^{16}$. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to $616,000$ by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity $s$ in any global function field of genus $g$ over ${\mathbb{F}}_q$ for $q\ge (1.14+0.16g)e^{6.5+0.97g}{s}^2$. In particular, there is no covering system of ${\mathbb{F}}_q[x]$ with distinct moduli for $q\ge 759$.

10 pages, accepted by J. Number Theory