2005/04/09 by Renling Jin, Jin, Renling
Computer Science · Mathematics · #03H15 #11B05 #11B13 #11U10 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Number Theory (math.NT) #math.LO #math.NT #msc:03H15 #msc:11B05 #msc:11B13 #msc:11U10
paper · pdf · doi:10.48550/arxiv.math/0504186
67 pages
arxiv created 2005/04/09 · openalex publication_date 2005/04/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.