### Abstract

We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that alpha and beta are positive reals, that N is a large prime and that C, D subset of or equal to Z/NZ have sizes gammaN and deltaN respectively. Then the sumset C + D contains an AP of length at least e(crootlog N), where c > 0 depends only on gamma and delta. In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of Z/NZ, and prove a structural result for sets with this property.

Translated title of the contribution | Arithmetic progressions in sumsets |
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Original language | English |

Pages (from-to) | 584 - 597 |

Journal | Geometric and Functional Analysis |

Volume | 12 (3) |

Publication status | Published - 2002 |

### Bibliographical note

Publisher: Birkhauser Verlag AgOther identifier: IDS Number: 587BT

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## Cite this

Green, BJ. (2002). Arithmetic progressions in sumsets.

*Geometric and Functional Analysis*,*12 (3)*, 584 - 597.