### 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) ≥ (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 sign_{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^{2K2-2}|A|.

Original language | English |
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Pages (from-to) | 43-52 |

Number of pages | 10 |

Journal | Bulletin of the London Mathematical Society |

Volume | 38 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2006 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Bulletin of the London Mathematical Society*,

*38*(1), 43-52. https://doi.org/10.1112/S0024609305018102