Arithmetic progressions in sumsets

We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that a alpha and b beta are positive reals, that N is a large prime and that C,D í Bbb Z/NBbb Z C,D subseteq {Bbb Z}/N{Bbb Z} have sizes gN gamma N and dN delta N respectively. Then the sumset C + D contains an AP of length at least e^{c ?{log} N} e^{c sqrt{rm log} N} , where c > 0 depends only on g gamma and d delta . In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of Bbb Z/NBbb Z {Bbb Z}/N{Bbb Z} , and prove a structural result for sets with this property.