Let $X$ be a random variable. $X$ is called lattice distributed if there exist real numbers $a, b$ such that $P(X \in a +b\mathbb{Z})=1$. Show that $X$ is lattice distributed if there exists $v\neq 0$ such that $\left | \varphi(v) \right |=1$. Here, $\varphi(t)=E\left[e^{itX}\right]$ denotes the characteristic function of $X$.
I have actually found a solution in Rick Durret's Probability: Theory and Examples, Theorem 3.5.1, but a major part of the proof is missing and I think he's using a theorem about strictly convex functions on a function that is convex, but not strictly convex. Anyway, there must be an easier solution to this I think.