On covering multiplicity

Abstract:

Let $A=\{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}$ be a system of arithmetic sequences which forms an $m$-cover of $\mathbb {Z}$ (i.e. every integer belongs at least to $m$ members of $A$). In this paper we show the following surprising properties of $A$: (a) For each $J\subseteq \{1,\cdots ,k\}$ there exist at least $m$ subsets $I$ of $\{1,\cdots ,k\}$ with $I\ne J$ such that $\sum _{s\in I}1/n_{s}-\sum _{s\in J}1/n_{s}\in \mathbb {Z}$. (b) If $A$ forms a minimal $m$-cover of $\mathbb {Z}$, then for any $t=1,\cdots ,k$ there is an $\alpha _{t}\in [0,1)$ such that for every $r=0,1,\cdots ,n_{t}-1$ there exists an $I\subseteq \{1,\cdots ,k\} \setminus \{t\}$ for which $[\sum _{s\in I}1/n_{s}]\geqslant m-1$ and $\{\sum _{s\in I}1/n_{s}\} =(\alpha _{t}+r)/n_{t}.$