= Uniform square recurrence on the circle
{title2=$\forall\epsilon>0\ \exists N\ \forall a\ \exists1\le m\le N:\|am^2\|<\epsilon$}
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> $am^2$ modulo one. Thus every parameter belongs to one of the open sets $\{a:\|am^2\|<\epsilon\}$. Compactness of the <circle group> gives a finite subcover, hence a bound $N$ uniform in $a$; no explicit rate follows from this compactness step.
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