On some problems in Additive number theory

Abstract

This thesis discusses some problems relating the properties of a set A and those of A+A, when A is a subset of an abelian group. Given a finite abelian group G and A is a subset of G, it is said that A is sum-free if the sets 2A and A are disjoint. Chapter 2 discusses the problem of finding the structure of all large sum-free subsets of G. The complete structure of all largest sum-free subsets of G, are obtained provided all the divisors of order G are congruent to 1 modulo 3. Also partial results are obtained regarding structure of all large maximal sum-free subsets of G. A sum-free set A is maximal if it is not a proper subset of any sum-free set. If there is a divisor of order of G which is not congruent to 1 modulo 3 then structure of all largest sum-free subsets of G was known before. The results in this thesis are based on a recent result of Ben Green and Imre Ruzsa. Chapter 3 improves the 'error term' in asymptotic formula of sigma (G) obtained by Ben Green and Imre Ruzsa, using slight refinement of the methods. Chapter 4 discusses a problem on an additive representation function, using an additive lemma proven by means of graph theory.