= Equidistribution criterion for a torus translation
{title2=$ku+\ell v\notin2\pi\mathbb Z$}
The sequence $r(u,v)$ is equidistributed in $\mathbb T^2$ precisely when $ku+\ell v\notin2\pi\mathbb Z$ for every nonzero integer pair $(k,\ell)$. Equivalently $1,u/(2\pi),v/(2\pi)$ are rationally independent. Under that condition, nonconstant character averages tend to zero by the geometric-series formula. Uniform approximation by <trigonometric polynomials> gives convergence for continuous functions, and continuous upper and lower bounds give convergence for rectangles. If a character is resonant, the orbit lies in its proper closed kernel and misses a positive-area rectangle, so equidistribution fails.
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