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A sharp result on $ m$-covers


Authors: Hao Pan and Zhi-Wei Sun
Journal: Proc. Amer. Math. Soc. 135 (2007), 3515-3520
MSC (2000): Primary 11B25; Secondary 11B75, 11D68, 11R04
DOI: https://doi.org/10.1090/S0002-9939-07-08890-9
Published electronically: August 15, 2007
MathSciNet review: 2336565
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Abstract: Let $ A=\{a_{s}+n_{s}\mathbb{Z} \}_{s=1}^{k}$ be a finite system of residue classes which forms an $ m$-cover of $ \mathbb{Z} $ (i.e., every integer belongs to at least $ m$ members of $ A$). In this paper we show the following sharp result: For any positive integers $ m_{1},\ldots ,m_{k}$ and $ \theta \in [0,1)$, if there is $ I\se \{1,\ldots ,k\}$ such that the fractional part of $ \sum _{s\in I} m_{s}/n_{s}$ is $ \theta $, then there are at least $ 2^{m}$ such subsets of $ \{1,\ldots ,k\}$. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to $ m$-covers of the integral ring of any algebraic number field with a power integral basis.


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Additional Information

Hao Pan
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: haopan79@yahoo.com.cn

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-07-08890-9
Received by editor(s): January 3, 2006
Received by editor(s) in revised form: June 3, 2006, and August 25, 2006
Published electronically: August 15, 2007
Additional Notes: The second author is responsible for communications and is supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) in China.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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