On the $j$-th smallest modulus of a covering system with distinct moduli

Abstract

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 $\le \exp (c{j}^2/\log (j+1))$.

8 pages, minor corrections and changes. Final version, to appear in Int. J. Number Theory