Green, Ben; Ruzsa, Imre Z. Sets with small sumset and rectification. (English) Zbl 1155.11307 Bull. Lond. Math. Soc. 38, No. 1, 43-52 (2006). Summary: We study the extent to which sets \(A\subseteq\mathbb Z/N\mathbb Z\), \(N\) 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 \(|A + A| \leq K|A|\) 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 obtain a reasonable bound: we can take \(c(K)\geq (32K)^{-12K^2}\). 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 \(A\subset\mathbb F_2^n\), and if \(|A + A| \leq K|A|\), then \(A\) is contained in a coset of a subspace of size no more than \(K^22^{2K^2-2} |A|\). Cited in 5 ReviewsCited in 35 Documents MSC: 11B75 Other combinatorial number theory PDFBibTeX XMLCite \textit{B. Green} and \textit{I. Z. Ruzsa}, Bull. Lond. Math. Soc. 38, No. 1, 43--52 (2006; Zbl 1155.11307) Full Text: DOI arXiv