Subsequence Sums Of Zero-sum Free Sequences

Additive theory is the study of sums of sets (or sequences). Additive theory is a main topic in combinatorial number theory. This theory has seen exciting developments and dramatic changes in direction in recent years, thanks to its connections with areas such as number theory, ergodic theory and graph theory.Based on the famous Kermperman-Sherk Theorem, we provide a lower bound for the size ofΣ(S) for a zero-sum free sequence S. All information on the minimal cardinality ofΣ(S) can be successfully applied to the investigation of a great variety of problems in combinatorial and additive number theory.In recent forty years, the study ofΣ(S) has attracted many researchers including R.B. Eggleton and P. Erdos, J.E. Olson, B. Bollobas and I. Leader, W.D. Gao, Y.O. Hamidoune, D.J. Grynkiewicz, S. Savchev and F. Chen, P.Z. Yuan, E. Balandraud, Dias da Silva, A. Pixton etc.In this thesis, the minimal cardinality ofΣ(S) for a zero-sum free sequence is our mam topic.This thesis consists of two parts. In Chapter 2, the first part of the thesis, contributes to the problems of subset sums. The author gives the result of F(6)= 19 which proves the conjecture proposed by R.B. Eggleton and P. Erdos in 1972. Then we will use the crucial new results for further progress on the study of the structure of long zero-sum free sequences.The second part, including Chapter 3 and Chapter 4, contributes to the problems of subsequence sums. In Chapter 3, we are interested in the conjecture proposed by Gao and Leader. We gives the result that if S is a zero-sum free sequence with |S|=exp(G) then f(S)≥2|S|-1.In Chapter 4, we continue to discuss the problems of subsequence sums and develop the theory of Gao and Leader. Let G be a finite abelian group. Suppose S is a zero-sum free sequence with |S|=exp(G)+1. We give a result of f(S)≥3exp(G)-1. Furthermore, we apply the result to the study of theΣ|G|(S).