Solution
= Solution
Set $Z=e^{iuX}$. Then $|Z|=1$ almost surely and
$$
\mathbb E\left[|Z-\mathbb EZ|^2\right]
=\mathbb E|Z|^2-|\mathbb EZ|^2
=1-|\widehat\mu_X(u)|^2=0.
$$
Thus $Z=\mathbb EZ$ almost surely. Its constant value has modulus one, so it equals $e^{i\theta}$ for some $\theta\in\mathbb R$. Consequently the <unit modulus of a characteristic function> criterion gives
$$
\boxed{\ uX\in\theta+2\pi\mathbb Z\quad\text{almost surely}.\ }
$$