Characterizing the structure of A when the ratio |2A|/|A| is bounded by 3+epsilon

Abstract

Let N be the set all of non-negative integers, let A be a finite subset of N, and let (2A) be the set of all numbers of form a+b for each a and b in A. The arithmetic structure of A was accurately characterized by Freiman when (i) |2A|<3|A|-3, (ii) |2A|=3|A|-3, or (iii) |2A|=3|A|-2. It is also suggested by Freiman that for characterizing the arithmetic structure of A when |2A|>3|A|-2, analytic methods need to be used. However, the interesting and more general results of Freiman, which use analytic methods, no longer give the arithmetic structure of A as precise as the results mentioned above. In this paper we characterize, with the help of nonstandard analysis, the arithmetic structure of A along the same lines as Freiman's results mentioned above when |2A|=3|A|-3+b where b is positive but not too large. Precisely, we prove that there is a positive real number epsilon and a natural number K such that if |A|>K and |2A|=3|A|-3+b for b between 0 and epsilon times |A|, then A is either a subset of an arithmetic progression of length at most 2|A|-1+2b or a subset of a bi-arithmetic progression of length at most |A|+b. The union of two arithmetic progressions I and J of the same difference d is called a bi-arithmetic progression if I+I, I+J, and J+J are pairwise disjoint.