Great Special Abel Group Generated Subset Of Integer Set Problems

In this paper,we use additive theory on the Abelian group to study two problems on the Abelian group,the direct problem,that is,when given A and B are two sets of Abelian group,what are properties and the structrure of the sumset A+B? and the inverse problem-when |A+B| is as small as possible,what are properties and the structure of the set A and the set B? And then we discuss some relative problems about intergers' sumsets.The main work of this paper has two aspects:observe Abelian group'sρ-extreme generating sets whet it has the form of(4,m).And get a non-trivial example of(G,ρ) that make t_ρ(G) =0.then try to spread a lemma and a theorem Vsevolod F.Lev got it.In a word,we researched some basic problems in additive group theory related to these.We discuss these problems in four chapters:Chapter 1 Introduction.We mainly introduce the problem of study on additive number theory and its background and development the main result of this paper.Chapter 2 We mainly investigated additive basis of Abelian group, and improve a G.Zemor's conclusion about additive basis.Chapter 3 Let G is an Abelian group in the form of(4,m),i.e. G=Z₄(?) Z_m,(4| m,m＞16 ),and then observe Abelian group'sρ- extreme generating sets whet it has the form of(4,m).And get a non-trivial example of(G,ρ) that make t_ρ(G) = 0.Chapter 4 We discuss some problems about sumset of integers.We mainly popularize a lemma and a theorem Vsevolod F.Lev got it.