## 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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