## Abstract

Let A be a flnite subset of an abelian group G. For every element b_{i} of the sumset 2A=b_{0}, b_{1},.., b_{2}A-1 we denote by D_{i}=a-a′. a, a′ϵA, a+a′=b_{i} and r_{i}=(a, a′). a+a′=b_{i}, a, a′ϵA. After an eventual reordering of 2A, we may assume that r0≥r_{1}≥..≥r_{2A}-1. For every 1≤s≤2A we deflne Rs(A) = D_{0}[D_{1}[..[D_{s-1} and Rs(k)= maxR_{s}(A). AG, A=kg. Bourgain and Katz and Tao obtained an estimate of R_{s}(k) assuming s being of order k. In this note we flnd the exact value of R_{s}(k) in cases s=1, s=2 and s=3. The case s=3 appeared to be not simple. The structure of extremal sets led us to sets isomorphic to planar sets having a rather unexpected form of a perfect hexagon. The proof suggests the way of dealing with the general case s≥4.

Original language | English |
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Pages (from-to) | 131-148 |

Number of pages | 18 |

Journal | Functiones et Approximatio, Commentarii Mathematici |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - 2007 |

## Keywords

- Inverse additive number theory
- Kakeya problem