= Solution
If $\beta<2^{-k}$, part (iii) gives the first alternative. Otherwise $\log(\beta^{-1})\le k\log2$, so part (ii) gives a subspace $V$ of codimension $O(\alpha^{-2}k)$ with
$$
\|1_A*\mu_V\|_2^2
=\sum_{t\in V^\perp}|\widehat{1_A}(t)|^2
\ge\frac{3\alpha^2}{2}.
$$
The function $1_A*\mu_V$ is nonnegative and has mean $\alpha$. Hence
$$
\|1_A*\mu_V\|_2^2
\le\|1_A*\mu_V\|_\infty\|1_A*\mu_V\|_1
=\alpha\|1_A*\mu_V\|_\infty.
$$
It follows that
$$
\boxed{\|1_A*\mu_V\|_\infty\ge3\alpha/2.}
$$
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