Zero Sum Problem On Free Abelian Groups Of Rank 2

Zero-sum theory is an important branch of combinatorial number theory.The main goal in zero-sum theory is to study the properties related to zero-sum sequences,especially minimal zero-sum sequences.Let G be an abelian group and X be a nonempty subset of G.If the sum of all terms in S is zero,then S is called zero-sum.If S is zero-sum and any proper nonempty subsequence of S is not zero-sum,then S is called minimal zero-sum.If S is minimal zero-sum and satisfies the following property:suppose a,b ? X,a+b=g E X and let T be the zero-sum sequence replacing a,b by g in S,then T is not minimal zero-sum,then S is called unsplittable minimal zero-sum sequence or unsplittable sequence in short.The number of terms in S is called the length of S.The Davenport of X is defined to be the supremum of the length of all minimal zero-sum sequences over X.Computing the Davenport constant and describing the structure of minimal zero-sum sequences are two core problems in zero-sum theory.The research of zero-sum problem over finite abelian groups has a long history over fifty years.There are fruitful and beautiful results in this area.However,few results are known for the zero-sum problems over infinite abelian groups.In the combinatorial point of view,the computation in the zero-sum problems is very complicated when the group has a high rank.The present paper mainly study the zero-sum problem over subset of free abelian group Z2.For any real number m ?n,let[m,n]={a ? Z:m?a?n} and let[m1,n1]×[m2,n2]={(a1,a2)?Z2,a1?[m1,n1],a2 ?[m2,n2]}.The main result of this paper are:In chapter 1,we introduce the background and the history of zero-sum theory.We also recall the terminology and notations in zero-sum theory and list some results about the structure of minimal zero-sum sequences over integers.In chapter 2,we mainly study the zero-sum problems over[-1,1]×[-m,n].We introduce the concept of minimal zero-sum sequence modulo an element.Let S be a minimal zero-sum sequence over[-1,1]×[-m,n].We fix a special factorization of S into minimal zero-sum sequences modulo(0,1).Combing with combinatorial technique and some known results,we prove that:for any positive integer m,n,the Davenport constant of[-1,1]×[-m,n]is 2(m+n).We also obtain the structure of minimal zero-sum sequences with maximal length 2(m+n).In chapter 3,we consider unsplittable sequences over[-1,n]×[-1,n],n? 4.Let S be a minimal zero-sum sequence over[-1,n]×[-1,n].We first give an estimate of the multiplicity vectors(-1,0),(0,-1),(-1,-1)based on the high non-symmetry of[-1,n]×[-1,n].Then we show that:if S is a unsplittable sequence over[-1,n]×[-1,n]with length no less that n2+n+2,then S is consisting of many copies of a and a minimal zero-sum sequence modulo ?,where?? {(-1,0),(0,-1),(-1,-1)}.In chapter 4,we consider the following question:given[-m1,n1]×[-m2,n2],mi,ni>0,fixed m1,n1,m2,the asymptotic behavior of the Davenport constant of[-m1,n1]×[-m2,n2]as n2??.We show that there exist a positive constant K,C independent of n2 such that D([-m1,n1]×[-m2,n2])?(m1+n1)n2+C,for n2>K.As an immediate consequence we have(?).