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We prove results about the L^p-almost-periodicity of convolutions. One ofthese follows from a simple but rather general lemma about approximating a sumof functions in L^p, and gives a very short proof of a theorem of Green that ifA and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B containsan arithmetic progression of length at least about exp(c (alpha beta logN)^{1/2}). Another almost-periodicity result improves this bound for densitiesdecreasing with N: we show that under the above hypotheses the sumset A+Bcontains an arithmetic progression of length at least about exp(c (alpha logN/(log(beta^{-1}))^3)^{1/2}).

Arithmetic progressions in sumsets and L^p-almost-periodicity