Research proof index

If a finite set $s$ is additively dissociated, then its order-$n$ additive energy is at most $C^n n^n |s|^n$, where $C$ is the project's Chang constant. This is the quantitative dissociated-set estimate used in the proof of Chang's lemma.

Chang's lemma for the large Fourier spectrum. If $f$ is nonzero and $\eta>0$, there is a subset $\Delta$ of the $\eta$-large spectrum such that the entire large spectrum lies in the additive span of $\Delta$. The theorem also gives the explicit bound $|\Delta| \le \left\lceil C e\,\left\lceil \mathcal L\!\left(\|f\|_1^2/(\|f\|_2^2|G|)\right)\right\rceil/\eta^2\right\rceil$, with the project's constant $C$.

A weighted Hölder lower bound for Fourier energy. If $\Delta$ lies in the $\eta$-large spectrum of $f$, $m\ne0$, and a weight $\nu$ is at least $1$ wherever $f$ is nonzero, then the order-$m$ energy of $\Delta$ weighted by $\widehat\nu$ is at least $|\Delta|^{2m}\eta^{2m}\|f\|_1^2/\|f\|_2^2$.