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Geometric & Functional Analysis GAFA

, Volume 12, Issue 3, pp 584–597 | Cite as

Arithmetic progressions in sumsets

  • B. Green

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 \( C,D \subseteq {\Bbb Z}/N{\Bbb Z} \) have sizes \( \gamma N \) and \( \delta N \) respectively. Then the sumset C + D contains an AP of length at least \( e^{c \sqrt{\rm log} N} \), 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 \( {\Bbb Z}/N{\Bbb Z} \), and prove a structural result for sets with this property.

Keywords

Structural Result Arithmetic Progression Positive Real 
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Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • B. Green
    • 1
  1. 1.Trinity College, Cambridge University, Cambridge CB2 1TQ, UKGB

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