On The Zero-sum Problems Over Product Of Integral Intervals Of Rank 2

Zero-sum theory is one of the branch of combinatorial number theory.Suppose that X is a nonempty subset of an abelian group,a sequence over X is called zero-sum,if the sum of all terms of the sequence is zero.If a sequence S is zero-sum and any nonempty subsequence of S is not zero-sum,then S is all minimal zero-sum.The Davenport constant of X,is defined to be the supremum of the length of all minimal zero-sum sequences over X.The main target of zero-sum theory is to describe the structure of zero-sum sequences and minimal zero-sum sequences and compute or estimate the Davenport constant.There is a long history of the study of zero-sum theory on finite abelian groups.However,there are not plenty results on the zero-sum problems on infinite groups.In the paper,we mainly investigate the structure of long minimal zero-sum sequences over subset[-1,n₁]×[-m₂,n₂]ofZ².We obtain several classes of long minimal zero-sum sequences.Under the condition n₂（?）n₁,m₂,we prove that the minimal zero-sum sequences over[-1,n₁]×[-m₂,n₂]with length no less than（n₁+1）（m₂+n₂）are exactly in these classes of minimal zero-sum sequences.As an immediate consequence,we show that the Davenport constant of[-1,n₁]×[-m₂,n₂]is exactly（n₁+1）（m₂+n₂）.