  In the present paper we prove a certain lemma about the structure of "lower
level-sets of convolutions", which are sets of the form $\{x \in \Z_N :
1_A*1_A(x) \leq \gamma N\}$ or of the form $\{x \in \Z_N : 1_A*1_A(x) < \gamma
N\}$, where $A$ is a subset of $\Z_N$. One result we prove using this lemma is
that if $|A| = \theta N$ and $|A+A| \leq (1-\eps) N$, $0 < \eps < 1$, then this
level-set contains an arithmetic progression of length at least $N^c$, $c =
c(\theta, \eps,\gamma) > 0$. It is perhaps possible to obtain such a result
using Green's arithmetic regularity lemma (in combination with some ideas of
Bourgain); however, our method of proof allows us to obtain non-tower-type
quantitative dependence between the constant $c$ and the parameters $\theta$
and $\eps$. For various reasons (discussed in the paper) one might think,
wrongly, that such results would only be possible for level-sets involving
triple and higher convolutions.
