On the structure of the spectrum of small sets

TL;DR

This paper investigates the structure of the spectrum of small subsets in finite abelian groups, providing new combinatorial results that extend understanding beyond previous trivial regimes and apply to very small sets.

Contribution

It demonstrates that for any set, a large subset exists whose spectrum's sumset has bounded size, extending combinatorial spectral analysis to smaller set sizes.

Findings

Existence of large subsets with bounded spectrum sumset

Results apply to sets of size as small as sub-constant regimes

Extension of spectral structure results beyond classical regimes

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 $A'$, 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.