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On Covering Multiplicity

Proceedings of the American Mathematical Society / v.127 no.5. 1999, pp.1293-1300

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Abstract : Let A = {a $_{s}$ + n $_{s}{\Bbb Z}$ } $_{s=1}^{k}$ be a system of arithmetic sequences which forms an m-cover of 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 $\neq $ J such that $\sum_{s\in I}$ 1/n $_{s}$ - $\sum_{s\in J}$ 1/n $_{s}\in {\Bbb Z}$ . (b) If A forms a minimal m-cover of Z, then for any t = 1, $\ldots $ ,k there is an $\alpha _{t}\in $ [0, 1) such that for every r = 0, 1, $\ldots $ , n $_{t}$ - 1 there exists an I $\subseteq $ {1, $\cdots $ , k} $\backslash $ {t} for which [ $\sum_{s\in I}$ 1/n $_{s}$ ] $\geq $ m - 1 and { $\sum_{s\in I}$ 1/n $_{s}$ }=( $\alpha _{t}$ + r)/n $_{t}$ .

Keyword : Primary 11B25 . Secondary 11A07, 11B75, 11D68

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