= Quantitative quadratic recurrence
{title2=$\|\alpha n^2\|<\varepsilon$}
For a universal constant $C>0$ and $0<\varepsilon\leq1$, every real $\alpha$ admits $1\leq n\leq\varepsilon^{-C}$ with <distance to the nearest integer> of $\alpha n^2$ less than $\varepsilon$. A <Fejér kernel> detects failure of recurrence as a large <quadratic exponential sum>. The <Van der Corput inequality for finite scalar sequences> then produces a short linear near-return, whose suitable multiple gives the quadratic return. To include $\varepsilon>1$, use the bound $\max\{1,\varepsilon^{-C}\}$.
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