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Let $X$ be a real random variable and $\phi$ its characteristic function. Show that there exist real $a,b$ such that $P\left(X \in a+b\mathbb{Z} \right) = 1$ if and only if there exists a nonzero $x$ such that $\left|\phi(x)\right| = 1$.

Any ideas or hints would be appreciated. Thanks.

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    Hint: How can the weighted average of points on a unit circle not end up in the interior of the circle?2012-12-04
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    Any attempt would be appreciated.2012-12-04

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