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

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 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 [Formula: see text] such that the [Formula: see text]th smallest modulus of a minimal covering system with distinct moduli is [Formula: see text].