当比率|2A| / |A|由3+ π限定时，表征A的结构
Characterizing the structure of A when the ratio |2A|/|A| is bounded by 3+epsilon
摘要：
设N是所有非负整数的集合，A是N的有限子集，（2A）是A中a和b的所有形式为a+b的数的集合.当（i）|2A| <3 |A|-3，（ii）|2A| =3 |A|-3，或（iii）|2A| =3 |A|-2时，Freiman准确地表征了A的算术结构。Freiman还建议，当|2A| >3 |A|-2时，为了表征A的算术结构，需要使用分析方法。然而，有趣的和更普遍的结果弗赖曼，其中使用的分析方法，不再给予算术结构的A作为精确的上述结果。本文利用非标准分析方法，刻划了当|2A| =3 |A|-3 +b时A的算术结构，其结果与Freiman的上述结果相同.精确地说，我们证明了存在一个正实数n和一个自然数K，使得如果|A| >K且|2A| =3 |A| -3+b，其中b在0和1/4之间乘以|A|，那么A是长度至多为2 |A| -1+2b的等差数列的子集或长度至多为|A| +b的双等差数列的子集.两个具有相同差d的等差数列I和J的并集称为双等差数列，如果I+I、I+J和J+J两两不相交。
英文摘要：
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.
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