W Inner one cft

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.

An $L^\infty$ almost-periodicity theorem. Under the small-growth hypothesis $\sigma[A,S]\le K$, and for nonempty finite sets $B,C$, there is a set of translations $T$ with $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil/\varepsilon^2}|S|$. Every $t\in T$ changes the normalized convolution $\mu_A*1_B*\mu_C$ by at most $\varepsilon$ in $L^\infty$.

A boosted $L^\infty$ almost-periodicity estimate. Under $\sigma[A,S]\le K$, it finds a large set $T$, with the stated lower bound $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil k^2/\varepsilon^2}|S|$, such that averaging the target convolution against the $k$-fold convolution of $\mu_T$ changes it by at most $\varepsilon$ in $L^\infty$.