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Source: arXiv v2, pp. 1 and 3, the opening definitions and Definition 2.2.

An arithmetic progression is a residue class $a+d\mathbb Z$, with positive integer modulus $d$. A finite family of such progressions is a covering system when its union is $\mathbb Z$. It is minimal when deleting any member destroys the covering property.

For an indexed finite family

$$\mathcal A=(a_i+d_i\mathbb Z)_{1\le i\le n},$$

its multiplicity is

$$m(\mathcal A)=\max_{d\ge1}\#\{i:d_i=d\}.$$

Thus a family has distinct moduli exactly when $m(\mathcal A)=1$. The indexed form matters for the shifted auxiliary family in Claim 2.1: two shifted occurrences may represent the same residue class, but they retain their indices and hence their stated multiplicity. Repeated occurrences do not change the union, and every subsequent sieve argument applies to an indexed finite family.

For a positive integer $n$, $P^+(n)$ denotes its largest prime factor, with $P^+(1)=1$, and $\omega(n)$ is the number of its distinct prime factors. Brackets $[d_1,\ldots,d_r]$ denote least common multiple. All logarithms in this source unit are natural logarithms.