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On covering multiplicity


Author: Zhi-Wei Sun
Journal: Proc. Amer. Math. Soc. 127 (1999), 1293-1300
MSC (1991): Primary 11B25; Secondary 11A07, 11B75, 11D68
DOI: https://doi.org/10.1090/S0002-9939-99-04817-0
Published electronically: January 27, 1999
MathSciNet review: 1486752
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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 $\{\sum _{s\in I}1/n_{s}\} =(\alpha _{t}+r)/n_{t}.$


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Additional Information

Zhi-Wei Sun
Email: zwsun@netra.nju.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-99-04817-0
Received by editor(s): August 13, 1997
Published electronically: January 27, 1999
Additional Notes: Supported by the National Natural Science Foundation of the People’s Republic of China and the Return-from-abroad Foundation of the Chinese Educational Committee.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society

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