On the structure of the spectrum of small sets

Abstract

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 $A^{\prime }$, such that the sumset of the spectrum of $A^{\prime }$ 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.