堆垒基与Narkiewicz常数的研究

Additive basis and the Narkiewicz constants are two important problems in combinatorial number theory. Combinatorial number theory is a new direction in combinatorics and number theory. Many mathematicians are interested in the direction now. These two problems interrelate. We study them combining the methods from number theory, combinatorics, algebra and probability. On the one hand we study them directly; on the other hand we study the inverse problems of them. First, we study the critical number of of finite groups and its iverse problem. Second, we concertrate on the Narkiewicz constants of finite abelian groups of rank two and related problems. For additive basis, we study it in probabilistic methods and the polynomial method. For the Narkiewicz constants, we establish a relation between the Narkiewicz constants and short zero-sum problems. So, we can study the poblem in a new method. Through the project we hope to obtain more results on these two problems and promote the development of the subject.

堆垒基与Narkiewicz常数的研究是组合数论中的两个重要课题。组合数论是当前组合数学与数论研究的热门方向。这两个课题相互联系，相互促进。本项目就是从直接问题和逆问题两个角度出发，综合利用数论、组合数学、代数以及概率的方法和技巧，对堆垒基与Narkiewicz常量进行研究。这包括在一般有限群上对 critical number及其逆问题的探索；非唯一分解理论中Narkiewicz常量及相关问题的探讨，尤其注重对秩为2的有限Abel群的Narkiewicz常数的研究。在研究堆垒基方面，我们发展了Alon的多项式方法并结合概率方法研究堆垒基及其逆问题；在Narkiewicz常数研究方面，我们建立了Narkiewicz常数研究与短零和理论之间的联系，为我们对Narkiewicz常数的研究提供了一个全新的研究视角。希望通过对此项目的探索，取得一些较有影响的研究成果，促进该学科的发展。