= Solution
Apply the <Croot-Sisask almost-periodicity theorem> with the sampling set $A$, $S=G$, $f=1_{-A}$, and error $\epsilon/k$. Since $A+G=G$, the doubling parameter is $K=\alpha^{-1}$, and $\|1_{-A}\|_p\le1$. We obtain $X\subseteq G$ with
$$
|X|\ge\alpha^{O(\epsilon^{-2}k^2p)}|G|
$$
such that
$$
\|\tau_t(1_{-A}*\mu_A)-1_{-A}*\mu_A\|_p\le\epsilon/k
$$
for every $t\in X-X$. Every $x\in kX-kX$ is a sum of $k$ elements of $X-X$. Telescoping these $k$ shifts and using translation invariance and the triangle inequality for the <Lp norm> gives
$$
\boxed{\|\tau_x(1_{-A}*\mu_A)-1_{-A}*\mu_A\|_p\le\epsilon.}
$$
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