The New Valuation Formula For The Frobenius Number Of The Vitek's Community, And Integer And Set Discussion

Many classical problems in additive group theory and additive number theory are direct problems,that is,let two subsets A and B of a group,what are properties and the structure of the sumset A+B? By contrast, in an inverse problem,when |A+B| is as small as possible,what are properties and the structure of the set A and the set B?The main work of this paper has three aspects. Firstly, we use additive theory to improve Vitek's bound and obtain a new estimated formulas for the Frobenius numbers; secondly, we introduce a new result of the direct problems on the Abelian group; lastly, we deduce computation on |2^A| in some special cases.This paper is organized as follows:Chapter 1 Introduction. We introduce mainly the problem of study on additive number theory, its backgroud and development and the main result of this paper.Chapter 2 A new estimated formulas for the Vitek's bound. We use mainly additive theory to obtain a new estimated formulas for the Frobenius numbers and a new Lemma. We improve Vitek's bound by the new estimated formulas for the Frobenius numbers and the new Lemma.Chapter 3 Some problems of the sumset on integers. We introduce (?)ainly a new result on the direct problems of the sumset on integers and we deduce computation on|2^A|in three special cases.