= Quadratic Weyl inequality
{title2=$|\sum_{n=0}^N e(\alpha n^2)|\ll((N^2/q+N+q)\log(2N))^{1/2}$}
For <coprime integers> $a,q$ with $1\le q\le Q$ and $|\alpha-a/q|\le1/(qQ)$, expand the squared <quadratic exponential sum> by its difference $h$. The <exponential geometric sum bound> gives $|S|^2\ll N+\sum_{h\le N}\min(N+1,\|2\alpha h\|^{-1})$. In each block of length at most $q/4$, the rotations $2\alpha h$ are separated by at least $1/(2q)$: the reduced denominator of $2a/q$ is at least $q/2$, and the approximation error between two block points is at most $1/(2q)$. The <separated reciprocal-distance sum> bounds each block by $O(N+q\log(2N))$. There are $O(N/q+1)$ blocks, giving the displayed inequality. For bounded $q$ the trivial estimate supplies the same conclusion.
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