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Sets with small sumset and rectification. (English) Zbl 1155.11307

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 \(K^22^{2K^2-2} |A|\).

MSC:

11B75 Other combinatorial number theory
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