Uniform square recurrence on the circle ($\forall ϵ>0\exists N\forall a\exists 1\le m\le N:\Vert a{m}^2\Vert <ϵ$)

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.