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

TL;DR

This paper investigates the size of the $j$-th smallest modulus in minimal covering systems with distinct moduli, establishing an upper bound that grows exponentially with respect to $j^2$ over the logarithm of $j+1$, building on recent foundational results.

Contribution

It provides a new upper bound on the $j$-th smallest modulus in minimal covering systems, extending previous work and demonstrating a specific growth rate.

Findings

The $j$-th smallest modulus is bounded above by an exponential function of $j^2/\log(j+1)$.

The result builds on recent simplified proofs of boundedness of the minimum modulus.

Establishes a universal constant $c$ for the growth rate of the $j$-th smallest modulus.

Abstract

Covering systems were introduced by Erd\H{o}s 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, Bollob\'as, 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 $\le \exp(cj^2/\log(j+1))$.