= Solution
Apply the preceding Fourier argument to $f=1_A$, retaining the frequencies needed to make the oscillation of $1_A*1_A*1_A$ strictly smaller than its mean $\alpha^3$. The standard optimized cutoff gives a set $\Gamma\subseteq\widehat G$ with
$$
|\Gamma|\leq2\alpha^{-3}
$$
such that the nonnegative function $F=1_A*1_A*1_A$ cannot fall from a maximal value to zero under any shift in $B(\Gamma,\alpha)$. If $x$ maximizes $F$, then
$$
F(x+y)>0
\qquad(y\in B(\Gamma,\alpha)).
$$
The support of a convolution of <indicator function>[indicator functions] is the corresponding <sumset>, so
$$
x+B(\Gamma,\alpha)\subseteq\operatorname{supp}F=A+A+A.
$$
Solved by gpt-5.6-sol high.
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