Bounds for Erd\H{o}s covering systems in global function fields

TL;DR

This paper extends the study of Erdős covering systems to global function fields, establishing bounds on the existence of such systems with certain properties and improving previous results in the area.

Contribution

It provides new bounds for covering systems in global function fields and demonstrates the non-existence of certain covering systems under specified conditions.

Findings

No covering system of multiplicity s exists in global function fields with genus g for large q.

No covering system with distinct moduli exists over _q[x] for q > 73.

Improves previous bounds and results on Erd
53s covering systems in function fields.

Abstract

Covering systems of the integers were introduced by Erd\H{o}s in 1950. Since then, many beautiful questions and conjectures about these objects have been posed. Most famously, Erd\H{o}s 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 Erd\H{o}s' 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.