## Abstract

Combining Freiman's theorem with Balog-Szemerédi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ℝ^{d} and ℤ^{d}.

Original language | English |
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Pages (from-to) | 327-341 |

Number of pages | 15 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 156 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2014 |

## ASJC Scopus subject areas

- General Mathematics

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