Arithmetic Progressions in Sumsets and Lp-Almost-Periodicity

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