6 Bohr sets
Let \(\nu : \widehat{G} \to \mathbb {R}\). The corresponding Bohr set is defined to be
The rank of \(\nu \), denoted by \(\mathrm{rk}(\nu )\), is defined to be the size of the set of those \(\gamma \in \widehat{G}\) such that \(\nu (\gamma ){\lt}2\).
(Basic API facts: Bohr sets are symmetric and contain \(0\). Also that, without loss of generality, we can assume \(\nu \) takes only values in \(\mathbb {R}_{\geq 0}\) - I think it might be easier to have the definition allow arbitrary real values, and then switch to non-negative only in proofs where convenient. Or could have the definition only allow non-negative valued functions in the first place.)
If \(\rho \in (0,1)\) and \(\nu :\widehat{G}\to \mathbb {R}\) then
There are at most \(\lceil 4/\rho \rceil \) many \(z_i\) such that if \(\lvert 1-w\rvert \leq \nu (\gamma )\) then \(\lvert z_i-w\rvert \leq \rho \nu (\gamma )/2\) for some \(i\). Let \(\Gamma =\{ \gamma : \nu (\gamma ) {\lt}2\} \) and define a function \(f:\mathrm{Bohr}(\nu )\to \lceil 2/\rho \rceil ^{\mathrm{rk}(\nu )}\) where for \(\gamma \in \Gamma \) we assign the \(\gamma \)-coordinate of \(f(x)\) as whichever \(j\) has \(\lvert z_j-\gamma (x)\rvert \leq \rho \nu (\gamma )/2\).
By the pigeonhole principle there must exist some \((j_1,\ldots ,j_{d})\) such that \(f^{-1}(j_1,\ldots ,j_d)\) has size at least \((\lceil 2/\rho \rceil )^{-\mathrm{rk}(\nu )}\left\lvert \mathrm{Bohr}(\nu )\right\rvert \). Call this set \(B'\). It must be non-empty, so fix some \(x\in B'\). We claim that \(B'-x\subseteq \left\lvert \mathrm{Bohr}(\rho \cdot \nu )\right\rvert \), which completes the proof.
Suppose that \(z=x+y\) with \(x,y\in B'\), and fix some \(\gamma \in \Gamma \). By assumption there is some \(z_j\in \mathbb {C}\) such that \(\lvert z_j-\gamma (x)\rvert \leq \rho \nu (\gamma )/2\) and \(\lvert z_j-\gamma (y)\rvert \leq \rho \nu (\gamma )/2\). Then by the triangle inequality,
and so \(z=y-x\in \mathrm{Bohr}(\rho \cdot \nu )\).
We say \(\nu :\widehat{G}\to \mathbb {R}\) is regular if, with \(d=\mathrm{rk}(\nu )\), for all \(\kappa \in \mathbb {R}\) with \(\left\lvert \kappa \right\rvert \leq 1/100d\) we have
For any \(\nu :\widehat{G}\to \mathbb {R}\) there exists \(\rho \in [\frac{1}{2},1]\) such that \(\rho \cdot \nu \) is regular.
To do.
If \(B\) is a regular Bohr set of rank \(d\) and \(\mu :G\to \mathbb {R}_{\geq 0}\) is supported on \(B_\rho \), with \(\rho \in (0,1)\), then
To do.
There is a constant \(c{\gt}0\) such that the following holds. Let \(B\) be a regular Bohr set of rank \(d\) and \(L\geq 1\) be any integer. If \(\nu :G\to \mathbb {R}_{\geq 0}\) is supported on \(L B_\rho \), where \(\rho \leq c/Ld\), and \(\lVert \nu \rVert _1=1\), then
To do.
There is a constant \(c{\gt}0\) such that the following holds. Let \(B\) be a regular Bohr set of rank \(d\), suppose \(A\subseteq B\) has density \(\alpha \), let \(\epsilon {\gt}0\), and suppose \(B',B''\subseteq B_\rho \) where \(\rho \leq c\alpha \epsilon /d\). Then either
there is some translate \(A'\) of \(A\) such that \(\left\lvert A'\cap B'\right\rvert \geq (1-\epsilon )\alpha \left\lvert B'\right\rvert \) and \(\left\lvert A'\cap B''\right\rvert \geq (1-\epsilon )\alpha \left\lvert B''\right\rvert \), or
\(\lVert 1_{A}\ast \mu _{B'}\rVert _\infty \geq (1+\epsilon /2)\alpha \), or
\(\lVert 1_{A}\ast \mu _{B''}\rVert _\infty \geq (1+\epsilon /2)\alpha \).
To do.