有限Abel群上的零和理论

Abel群上的零和问题是近些年来备受人们关注的属于组合数论的一个研究方向。本项目旨在深化零和理论研究。我们将侧重研究如下几方面内容: (1)研究近年来由本申请人等提出的一般性零和不变量(定义见正文); (2) 研究零和理论中最重要的不变量之一Davenport 常数。对秩为3的群确定该常数，以及对一般的有限Abel群给出其更好的上界估计；(3) 研究关于具有大的幂指数的有限Abel群上的零和问题；(4) 运用概率方法，分析方法及代数方法研究零和问题。

Zero-sum problems has been attracted a lot of attention in recent years. In this project we aim to make some further and deeper study on zero-sum theory. We shall focus on the following: (1) to study zero-sum invariants in a general view which was first formulated in an recent paper by the applicant and others; (2) to determine the exact values of Davenport constants for finite abelian groups of rank three and give better upper bounds for Davenport constant for all finite abelian groups; (3) to study zero-sum problems on finite abelian groups of large exponents; and (4) to study some zero-sum problems by using probabilistic method, analytical menthod and algebraic method .