An upper bound for the minimum modulus in a covering system with squarefree moduli

TL;DR

This paper establishes an upper bound of 118 for the smallest modulus in covering systems with distinct squarefree moduli, and shows the k-th smallest modulus is bounded by an absolute constant.

Contribution

It provides the first explicit upper bound for the minimum modulus in such covering systems with squarefree moduli.

Findings

Minimum modulus in covering systems with squarefree moduli is at most 118.

The k-th smallest modulus in any covering system with distinct moduli is bounded by an absolute constant.

The results extend previous work by providing explicit bounds.

Abstract

Based on work of P. Balister, B. Bollob\'as, R. Morris, J. Sahasrabudhe and M. Tiba, we show that if a covering system has distinct squarefree moduli, then the minimum modulus is at most 118. We also show that in general the $k^{\rm th}$ smallest modulus in a covering system with distinct moduli (provided it is required for the covering) is bounded by an absolute constant.