Structure of Sets with Small Sumset and Applications

The famous (3k ¿ 4)-Theorem of Freiman states that if the doubling A+A of a set A of coprime integers satis¿es A + A ¿ 3|A| ¿ 4, then A is an interval with at most |A| ¿ 3 holes. It occurs, that sets with the same number of holes do not necessarily have doublings of the same size. It depends on the position of the holes. It was the main objective of this master thesis to determine the position of the holes for sets with small doubling. The answer to this question was given recently by Freiman and in here it is generalized to the case with di¿erent summands. In the main result, it is proved that if A and B are sets with same diameter and small sumset, then A + B contains an interval of length at least half the total length of A + B. If x is a hole of A + B in the left of the interval, then x is a hole of both A an B, and if it is a hole at the right side of the interval, then x ¿ l is a hole of A and B. Applications of this results are also presented, concerning difference sets, sum-free sets and the Frobenius problem.