On Erdős covering systems in global function fields

Pub Date : 2024-08-20 DOI:10.1016/j.jnt.2024.07.002

Huixi Li , Biao Wang , Chunlin Wang , Shaoyun Yi

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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 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 $F_{q}$ for $q \geq (1.14 + 0.16 g) e^{6.5 + 0.97 g} s^{2}$ . In particular, there is no covering system of $F_{q} [x]$ with distinct moduli for $q \geq 759$ .

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论全局函数域中的厄尔多斯覆盖系统

整数的覆盖系统是算术级数的有限集合，其联合是整数集合。关于覆盖系统的一个著名问题是厄尔多斯在 1950 年提出的最小模问题，他问在这种具有不同模的系统中，最小模是否可以任意大。2015 年，霍夫解决了这一问题，他证明了最小模量最多为 1016。2022 年，Balister、Bollobás、Morris、Sahasrabudhe 和 Tiba 通过发展 Hough 方法，将 Hough 的界限降低到 616,000。他们称之为扭曲法。在本文中，通过应用这一方法，我们主要证明了在 Fq 上任何属 g 的全局函数域中，不存在任何乘数为 s 的覆盖系统，即 q≥(1.14+0.16g)e6.5+0.97gs2。特别是，在 q≥759 时，Fq[x]不存在具有不同模数的覆盖系统。

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