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