On the jth smallest modulus of a covering system with distinct moduli

Covering systems were introduced by Erdos 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 et al. 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 jth smallest modulus of a minimal covering system with distinct moduli is <= exp(cj(2)/log(j + 1)).