Bounds for Erdős covering systems in global function fields

Abstract

Covering systems of the integers were introduced by Erdős in 1950. Since then, many beautiful questions and conjectures about these objects have been posed. Most famously, Erdős asked whether the minimum modulus of a covering system with distinct moduli is arbitrarily large. This problem was resolved in 2015 by Hough, who proved that the minimum modulus is bounded. In 2022, Balister et al. developed Hough's method and gave a simpler but more versatile proof of Hough's result. Their technique has many applications in a number of variants on Erdős' minimum modulus problem. In this paper, we show that there is no covering system of multiplicity $s$ in any global function field of genus $g$ over $\mathbb{F}_q$ for $q\geq(82.26+18.88g)e^{0.95g}s^2$. Moreover, we obtain that there is no covering system of $\mathbb{F}_q[x]$ with distinct moduli for $q>73$. This improves the results in the previous work of the author joint with Li, Wang and Yi.