Zero-Sum Subsequences Of Distinct Lengths And Representation Of Zero-Sum Invariants By Sets Of Zero-Sum Sequences

Zero-sum theory is an important branch of combinatorial number theory and it has developed rapidly and received extensive attention in recent years.One of the ba-sic research topics of zero-sum theory is to study the existence conditions of zero-sum subsequences with prescribed properties.From this,many invariants on finite abelian groups have been proposed,such as EGZ-constant,Davenport constant,η-constant and so on.In the 1970s,R.Graham posed a problem of investigating zero-sum subsequences with distinct lengths.He conjectured that a sequence over a cyclic group of order p with length p must contain two nonempty zero-sum subsequences of distinct lengths if the three terms are pairwise different,where p is a prime.P.Erd（?）s and E.Szemerédi proved the above conjecture when p is a sufficiently large prime.Then in 2010,Gao,Hamidoune and Wang extended the above conjecture to arbitrary positive integer n.In2012,B.Girard proposed a natural problem of determining the smallest integer t such that every sequence S over G of length|S|≥t has two nonempty zero-sum subsequences of distinct lengths,which is denoted by disc（G）.Recently,we study the inverse problem of disc（G）.Define L1（G）to be the set of all positive integers t with the property that there is a sequence S over G with length|S|=disc（G）-1 such that all nonempty zero-sum subsequences of S have the same length t.We study the determination of L1（G）.Related to disc（G）,Gao,Li,Peng and Wang[1]defined q′（G）to be the smallest integer t such that every sequence S over G of length|S|≥t has two nonempty zero-sum subsequences,say T1,T2,with vg（T1）≠vg（T2）for some g∈G.In order to describe the zero-sum invariants uniformly,Gao et al.[1]proposed the method of using sets of the zero-sum sequences to represent the zero-sum invariants.Let B（G）denote the set of all zero-sum sequences over G.For_?（?）B（G）,let d_?（G）be the smallest integer t such that every sequence S over G of length|S|≥t has a subsequence in_?.A zero-sum sequence S is essential with respect to some t≥D（G）if every_?（?）B（G）with d_?（G）=t contains S.Thus a natural research problem is to determine the smallest integer t such that there is no essential zero-sum sequence with respect to t,denote it by q（G）.A sequence S over G is a weak-regular sequence if vg（S）≤ord（g）for every g∈G.In this thesis,we give a new combinatorial constant,define N（G）as the smallest integer t such that every weak-regular sequence S over G of length|S|≥t has a nonempty zero-sum subsequence T of S satisfying that vg（T）=vg（S）>0 for some g|S.In this thesis,we first pay our attention to disc（G）and L1（G）on the groups G=C2⊕C2m⊕C2mnwith m,n∈N.Then we make some further study on d_?（G）and get some new results.This thesis is organized as follows.In Chapter 1,we introduce the definitions,notation,and background of our re-search problems.We also list our main results of this thesis.In Chapter 2,we determine the precise value of disc（G）on the groups G=C2⊕C2m⊕C2mnwith m,n∈N.In addition,we study the inverse problem L1（G）of disc（G）,and prove that L1（G）={exp（G）}.In Chapter 3,we investigate N（G）on finite abelian groups.We determine the exact value of N（G）for the cyclic groups of prime power order and the elementary abelian p-groups of rank two.In Chapter 4,we consider d_?（G）and its related invariants on finite abelian groups.And we prove that q（G）=q′（G）for any finite abelian group G.