= Translated Bohr neighborhood in a triple sumset
{title2=$t_0+B(\Gamma,\alpha/4)\subseteq A+A+A,\quad |\Gamma|\leq64\alpha^{-3}$}
For a subset $A$ of density $\alpha$ in a finite cyclic group, put $F=1_A*1_A*1_A$ using <normalized convolution on a finite group>. Its average is $\alpha^3$, so choose $t_0$ with $F(t_0)\geq\alpha^3$. Retain the <large spectrum> where $|\widehat{1_A}|\geq\alpha^2/8$. The <Parseval identity on a finite group> gives at most $64\alpha^{-3}$ frequencies. Their phases change by at most $\alpha/4$ on the associated <Bohr set>, contributing at most $\alpha^3/4$ to the change of $F$. The discarded frequencies contribute at most another $\alpha^3/4$, since $\sum |\widehat{1_A}|^2=\alpha$. Therefore $F(t_0+b)\geq\alpha^3/2>0$ on the <Bohr set>, proving the containment.
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