Arithmetic Progressions Almost Periodicity

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 $L^\infty$.

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 $\varepsilon$.

For a finite ambient group, the chord model of a Bohr set $B$ is contained in the arc model after widening $B$ by the factor $\pi/2$: $B_{\mathrm{chord}}\subseteq ((\pi/2)B)_{\mathrm{arc}}$.

Characterization of the order on Bohr sets. The relation $B_1\le B_2$ holds exactly when every frequency of $B_2$ is also a frequency of $B_1$, and $\operatorname{width}_{B_1}(\psi)\le \operatorname{width}_{B_2}(\psi)$ for each such frequency.

A point $x$ belongs to the arc model of a Bohr set $B$ exactly when every frequency $\psi$ of $B$ satisfies $\|\operatorname{angle}(\psi(x),1)\|\le \operatorname{width}_B(\psi)$. Thus membership can be checked using only the stored frequencies and widths.

A point $x$ belongs to the chord model of a Bohr set $B$ exactly when $\|1-\psi(x)\|\le \operatorname{width}_B(\psi)$ for every frequency $\psi$ of $B$.