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16 July 2020
 
  » arxiv » math.NT/0305369

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A Unified Theory of Zero-sum Problems, Subset Sums and Covers of Z
Zhi-Wei Sun ;
Date 27 May 2003
Subject Number Theory; Combinatorics MSC-class: 11B75; 05A05; 05C07; 11B25; 11C08; 11D68; 11P70; 11T99; 20D60 | math.NT math.CO
AbstractZero-sum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some connections among these seemingly unrelated fascinating areas. In this paper we establish the surprising connections for the first time and present a further unified approach. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorems in this paper unify many results in the three realms and also have applications in finite fields and graph theory.
Source arXiv, math.NT/0305369
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