Equidistributed sequence

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is equidistributed if the limiting frequency in every half-open interval equals that interval's length. Equivalently, its empirical probability measures converge to normalized Lebesgue measure when tested against every continuous function.

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.

Interval discrepancy measures the difference between empirical counts and the expected count for normalized Lebesgue measure. Convolving an enlarged interval indicator function with a narrow unit-integral indicator function gives a continuous majorant. Its Fourier coefficients decay as $O(1/(\delta k^2))$, converting the count into a constant term, finite exponential sums and a controlled absolute tail.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is an equidistributed sequence if and only if $N^{-1}{\sum }_{n=0}^{N-1}{e}^{2\pi im{x}_n}\to 0$ for every nonzero integer $m$. Approximation of continuous functions by trigonometric polynomials proves the criterion.

If a circle-valued sequence is not an equidistributed sequence, some positive-shift difference is not equidistributed either. The contrapositive follows by applying the Van der Corput inequality for finite scalar sequences to every nonzero integer exponential sum. Thus cancellation for all fixed nonzero differences implies equidistribution of the original sequence.

If $a$ is irrational and $b,c$ are real, the fractional parts of $a{n}^2+bn+c$ form an equidistributed sequence. In the Van der Corput lemma, every nonzero-lag correlation of $e^{2\pi im(a{n}^2+bn+c)}$ is a geometric series with irrational frequency $2mah$ and tends to zero. The Weyl criterion then applies.

For rational $a$, a multiple of its denominator gives exact integer recurrence. For irrational $a$, Weyl differencing and the Weyl criterion give an equidistributed sequence $a{m}^2$ modulo one. Thus every parameter belongs to one of the open sets $\{a:\Vert a{m}^2\Vert <ϵ\}$. Compactness of the circle group gives a finite subcover, hence a bound $N$ uniform in $a$; no explicit rate follows from this compactness step.