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13 July 2020
 
  » arxiv » math.NT/0504413

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A sharp result on m-covers
Hao Pan ; Zhi-Wei Sun ;
Date 20 Apr 2005
Subject Number Theory; Combinatorics MSC-class: 11B25; 11B75; 11D68; 11R04 | math.NT math.CO
AbstractLet A={a_s+n_sZ}_{s=1}^k be a finite 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 sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,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,...,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 a suitable algebraic number field.
Source arXiv, math.NT/0504413
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