论文标题
在$ j $ - 带有不同模块的覆盖系统的最小模量
On the $j$-th smallest modulus of a covering system with distinct moduli
论文作者
论文摘要
ErdőS于1950年引入了覆盖系统。在他介绍它们的同一篇文章中，他询问了具有不同模量的覆盖系统的最小模量是否有界限。 2015年，霍夫肯定回答了这个长期的问题。在2022年，Balister，Bollobás，Morris，Sahasrabudhe和Tiba提供了更简单，更广泛的霍夫结果证明。在他们的工作的基础上，我们证明存在一些绝对常数$ c> 0 $，因此，$ j $ - 最小覆盖系统的最小覆盖系统的最小模量为$ \ le \ exp（cj^2/\ log（j+1））$。
Covering systems were introduced by Erdős in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba gave a simpler and more versatile proof of Hough's result. Building upon their work, we show that there exists some absolute constant $c>0$ such that the $j$-th smallest modulus of a minimal covering system with distinct moduli is $\le \exp(cj^2/\log(j+1))$.
