= Solution
Let $T=1_A*1_A$ and choose
$$
m=\max\{1,\lceil\log(1/\alpha)\rceil\}.
$$
Apply part b with its error parameter replaced by a sufficiently small absolute multiple of $\varepsilon$. This produces a <vector subspace> $V$ of codimension
$$
O(m\varepsilon^{-2})=O(\varepsilon^{-2}\log(1/\alpha))
$$
with the evident harmless modification when $\alpha=1$.
For $t\in V$, $x\in\mathbb F_p^n$, and $q=2m$, <Hölder's inequality> gives
$$
\begin{aligned}
|(T*1_A)(x+t)-(T*1_A)(x)|
&\leq\sum_{a\in A}|T(x+t-a)-T(x-a)|\\
&\leq |A|^{1-1/q}\|\tau_tT-T\|_q.
\end{aligned}
$$
Since $\alpha^{-1/q}\leq e^{1/2}$ by the choice of $m$, the bound from part b is at most
$$
\varepsilon |A|^{2-1/q}N^{1/q}
=\varepsilon\alpha^{-1/q}|A|^2
\leq\varepsilon|A|^2
$$
after absorbing the absolute factor into the chosen error parameter. This is the required uniform estimate for $1_A*1_A*1_A$.
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