Result: On covering multiplicity

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