On the structure of the spectrum of small sets

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 $|A| = |G|^α$ whenever $α\le c$, 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 $γ_1,γ_2$ in the spectrum, $γ_1+γ_2$ 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 $|A| = |G|^α$ for any constant $α>0$, and even in some sub-constant regime.

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