Irrational unit-modulus frequencies force a point mass

If $u/v$ is irrational and $|{\phi }_X(u)|=|{\phi }_X(v)|=1$, then $X$ is almost surely constant. For independent copies $X,X^{\prime }$, both $u(X-X^{\prime })$ and $v(X-X^{\prime })$ lie in $2\pi \mathbb{Z}$ almost surely; irrationality makes the two corresponding lattices intersect only at zero, so $X=X^{\prime }$ almost surely. Two independent identically distributed variables can agree almost surely only when their common law is a point mass.