2004/03/21 by Ben Green, Green, Ben, Imre Z. Ruzsa +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.math/0403338
9 pages, minor corrections made
openalex publication_date 2004/03/21 · arxiv created 2005/01/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the extent to which sets A in Z/NZ, N prime, resemble sets of integers from the additive point of view (``up to Freiman isomorphism''). We give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and |A| < c(K)N then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we get a reasonable bound: we can take c(K) > exp(-cK2 log K). As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example if A is a subset of F2n, and if |A + A| < K|A|, then A is contained in a coset of a subspace of size no more than 2CK2|A|.