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.
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Submitted: August 2001.
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Green, B. Arithmetic progressions in sumsets . GAFA, Geom. funct. anal. 12, 584–597 (2002). https://doi.org/10.1007/s00039-002-8258-4
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DOI: https://doi.org/10.1007/s00039-002-8258-4