2024/02/06 by Huixi Li, Biao Wang, Li, Huixi +5
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2402.03810
openalex publication_date 2024/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 1016. 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 \mathbbFq for q≥ (1.14+0.16g)e6.5+0.97gs2. In particular, there is no covering system of \mathbbFq[x] with distinct moduli for q≥ 759.