  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.
