2013/08/03 by Renling Jin, Jin, Renling · 1 citation
Economics, Econometrics and Finance · Mathematics · #11P70 #Advanced Topology and Set Theory #FOS: Mathematics #Italy: Economic History and Contemporary Issues #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11P70
paper · pdf · doi:10.48550/arxiv.1308.0741
arxiv created 2013/08/03 · openalex publication_date 2013/08/03 · arxiv updated 2013/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a finite set of integers. We prove that if |A| is at least 2 and |A+A| is 3|A|-3, then one of the following is true: 1. A is a bi-arithmetic progression; 2. A+A contains an arithmetic progression of length 2|A|-1; 3. |A| is 6 and A is Freiman isomorphic to the set (0,0),(0,1),(0,2),(1,0),(1,1),(2,0); 4. A is Freiman isomorphic to a set in either the form of 0,2,...,2k union B union n for some non-negative integer k at most n/2 -2 or the form of 0 union C union D union n, where n=2|A|-2, B is left dense in [2k,n-1], C is right dense in [1,u] for some u in [4,n-6], D is left dense in [u+2,n-1], B,C,D are anti-symmetric and additively minimal in the correspondent host intervals.