Equidistribution of a quadratic polynomial with irrational leading coefficient

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.