Sets with small sumset and rectification

Ben Green, Imre Z. Ruzsa

Research output: Contribution to journalArticle

34 Citations (Scopus)


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) ≥ (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 ⊆ double struck F sign2n, and if |A + A| ≤ K|A|, then A is contained in a coset of a subspace of size no more than K222K2-2|A|.

Original languageEnglish
Pages (from-to)43-52
Number of pages10
JournalBulletin of the London Mathematical Society
Issue number1
Publication statusPublished - Feb 2006

ASJC Scopus subject areas

  • Mathematics(all)

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