Sets with small sumset and rectification

We study the extent to which sets A subset of Z/NZ, AF 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 vertical bar A + A vertical bar <= K vertical bar A vertical bar and vertical bar A vertical bar < c(K)N, then A is Freiman isomorphic to a set of integers. Because we avoid appealing to 2 Freiman's structure theorem, we obtain a reasonable bound: we can take c(K) >= (32K)(-12K2). 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 of F-2(n), and if vertical bar A + A vertical bar <= K vertical bar A vertical bar, then A is contained in a coset of a subspace of size no more than K(2)2(2K2-2)vertical bar A vertical bar.