  Let $G$ be a finite abelian group and $A$ a subset of $G$. The spectrum of
$A$ is the set of its large Fourier coefficients. Known combinatorial results
on the structure of spectrum, such as Chang's theorem, become trivial in the
regime $|A| = |G|^\alpha $ whenever $\alpha \le c$, where $c \ge 1/2$ is some
absolute constant. On the other hand, there are statistical results, which
apply only to a noticeable fraction of the elements, which give nontrivial
bounds even to much smaller sets. One such theorem (due to Bourgain) goes as
follows. For a noticeable fraction of pairs $\gamma_1,\gamma_2 $ in the
spectrum, $\gamma_1+\gamma_2$ belongs to the spectrum of the same set with a
smaller threshold. Here we show that this result can be made combinatorial by
restricting to a large subset. That is, we show that for any set $A$ there
exists a large subset $A'$, such that the sumset of the spectrum of $A'$ has
bounded size. Our results apply to sets of size $|A| = |G|^{\alpha}$ for any
constant $\alpha>0$, and even in some sub-constant regime.
