| Many studies have been done on the structure and the cardinality of sum of sets. In particular, let h be a positive integer, and let A be a finite subset of Zn. We are interested in hA, the h-fold sumset of A and its structure and cardinality for h sufficiently large. In this research, we are interested in extending some of the concepts on hA to h1 A1 + ··· + hrA r, where A1,...,A r, are finite subsets of Zn and h1,...,hr are positive integers.; It was found by Nathanson that when A is a set of integers, the structure of the h-fold sumset of A consists of an interval of consecutive integers and the cardinality of hA is a linear function of h. If we consider A to be a finite set of lattice points in Z n, Khovanskii has found a polytope in R n such that hA contains all of the lattice points in the polytope. Khovanskii found the cardinality of hA to be a function of hn.; The objective of this paper is three-fold. In Chapter 1, we let A1,...,Ar be finite subsets of integers, and let h1,...,h r be positive integers. We are able to generalize Nathanson's theorem to a sum of sumsets h1A 1 +···+ hrAr and determine the structure and estimate the cardinality of h 1A1 +···+ hrAr for all sufficiently large integers hi.; In Chapter 2, the author will present a modified proof of Khovanskii's theorems concerning hA. Furthermore, we let A 1,...,Ar be finite subsets of Zn, and h1,..., hr be positive integers, we are able to generalize Khovanskii's theorem to h1A1+···+ hrAr and determine its structure for hi sufficiently large. We are also able to estimate the cardinality of the linear form in the case of Z2.; In Chapter 3, we look at specifically the fine structure of the h-fold sumset of a set A in Z 2. It is found that the distribution of hA in the boundary region of the convex hull has a consistent regular pattern. To study the distribution of hA in the boundary region of the convex hull of hA, we partition the boundary region into many smaller regions and find that the distribution of hA in each small boundary region is identical, the elements from one region and the elements from another region differ by a translation. |
