Add Dissociated boring Energy le

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 $L^\infty$.

A finite-field approximation lemma. If $A_1$ and $A_2$ each have density at least $\alpha$, then for any test set $S$ and $0<\varepsilon\le1$ there is a subspace $V$ of explicitly bounded codimension such that smoothing $\mu_{A_1}*\mu_{A_2}$ by the uniform measure on $V$ changes its total mass on $S$ by at most $\varepsilon$.