= Bohr-set almost periodicity of a convolution
{c}
{title2=$\|f-T_uf\|_2^2\leq\varepsilon^2\alpha^3+4\theta^2\alpha$}
Let $A\subseteq\mathbb Z_n$ have <density of a finite subset> $\alpha$ and let $f=\mathbf1_A*\mathbf1_A$ use <normalized convolution on a finite group>. If $K=\{r:|\widehat{\mathbf1_A}(r)|\geq\theta\}$ and $u\in B(K;\varepsilon)$, then the displayed estimate holds for normalized physical-space <L2 norm> and $T_uf(x)=f(x-u)$. The <Bohr set> controls the translation factors on the <large spectrum>; the <fourth Fourier moment bound for an indicator function> controls their total weight. Outside $K$, use $|1-e^{2\pi iru/n}|\leq2$ and <Parseval identity on a finite group>.
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