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 about exp(c (alpha beta log N)^{1/2}). 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 about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}).
Long Arithmetic Progressions in Sets with Small Sumset
设$A, B\subseteq \mathbb{Z}$是有限的非空子集，且具有$\min A=\min B=0$，令$ $δ(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $ \max B \leq \max A \leq |A| + |B|-3 $ and \label{one}|A+B|\leq |A|+2|B|-3-δ(A,B), then we show $ A+B $ contains an arithmetic progression with difference 1 and length $ |A| + |B|-1 $. As a corollary, if \eqref{one} holds, $ \max（B）\leq \max（A）$ and either $ \gcd（A）= 1 $ or else $ \gcd（A+B）=1 $ and $| A+B| \leq 2 |A| + |B| -3 $, then $ A+B $ contains an arithmetic progression with difference 1 and length $ |A| + |B|-1 $.
