On exactly \(m\) times covers (Q1802726)

Let \(a+n\mathbb{Z}\) denote the arithmetic sequence \(\{x\in \mathbb{Z}: x\equiv a\pmod n\}\). In this paper, the author proves that if every integer is covered by \(a_ i = n_ i\mathbb{Z}\) \((i = 1,2,\dots,k)\) exactly \(m\) times, then for each \(n = 1,\dots,m\), there exist at least \(m\choose n\) subsets \(I\) of \(\{1,\dots,k\}\) such that \(\sum_{i \in I}1/n_ i\) equals \(n\). The bound \(m\choose n\) is best possible.