 Equidistribution criterion for a torus translation

The sequence $r(u,v)$ is equidistributed in ${\mathbb{T}}^2$ precisely when $ku+\ell v\notin 2\pi \mathbb{Z}$ 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.