Lattice distributed random variable and its characteristic function

Question

I'm working on the following problem and I''m stuck at the begining.

Let $X$ be a lattice distributed random variable, i.e $\exists\, a, b \in \mathbb{R}$ s.t. $$P[X \in a + b\mathbb{Z}]=1$$

Let $\varphi$ be the characterstic function of $X$.

Show that if $X$ is lattice - distributed, then there exists $u\neq 0$ such that $$\lvert\varphi(u)\rvert = 1$$

I first noticed that: $\lvert e^{itX}\rvert = 1$ Which could mean that:

$\lvert\varphi(u)\rvert = \lvert e^{itX}\rvert = 1$

Then:

$\varphi(u) = E[e^{iuX}] = e^{itX}$ for some $t\in\mathbb{R}$.

Any help would be appreciated.

Answer 1

If $b=0$ the any $u$ will do. Suppose $b \neq 0$. Take $u=2\pi /b$. We have $\phi (u)=Ee^{(2i\pi/b) X}=\sum e^{2i \pi (\frac a b+n)} P(X=a+b n)=e^{2i \pi (\frac a b)}$ so $|\phi (u)|=1$.