A group-theoretic approach to covering systems (Q2798883)

A covering (system) of the integers is a finite system of congruences \(x\equiv r_i\pmod{n_i}\) for \(1\leq i\leq t\), such that every integer satisfies at least one of these congruences. The authors demonstrate a possibility how to impose an algebraic structure on the set of all coverings with a fixed set of moduli to describe relationships among the elements of the set \(\Gamma_M\) of all coverings having moduli in a finite set of moduli \(M\). In the center of their algebraic approach is the group action of the holomorph \({\mathcal{G}}=\Aut({\mathbb Z}_L)\ltimes{\mathbb Z}_L)\), where \({\mathbb Z}_L\) is the additive group of integer modulo \(L=\text{lcm}(M)\). For instance, there is a natural (left) action of \({\mathcal G}\) on \(\Gamma_M\) (Theorem 4.3). This action is used to enumerate and categorize the elements of \(\Gamma_M\).