Arithmetic Progressions in Sumsets and L-almost-periodicity

We prove results about the L-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L, and gives a very short proof of a theorem of Green that if A and B are subsets of {1, . . . , N} of sizes αN and βN then A + B contains an arithmetic progression of length at least exp ( c(αβ logN) − log logN ) . 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 ( α logN log 2β−1 )1/2 − log(β−1 logN) ) .