SETS WITH SMALL SUMSET AND RECTIFICATION

Abstract

We study the extent to which sets A ⊆ 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 obtain a reasonable bound: we can take c ( K ) ⩾ ( 32 K ) − 12 K 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 ⊆ F 2 n , and if |A + A| ⩽ K|A|, then A is contained in a coset of a subspace of size no more than K 2 2 2 K 2 − 2 | A | . 2000 Mathematics Subject Classification 11B75.