= Badly approximable number
{title2=$\inf_{q\geq1}q\|q\alpha\|>0$}
An irrational real number is badly approximable when $\|q\alpha\|\geq c/q$ for some $c>0$ and every positive integer $q$, where the double bars mean distance to the nearest integer. Equivalently every rational approximation obeys $|\alpha-a/q|\geq c/q^2$. The <Diophantine bound for the square root of two> gives a concrete example and supplies separated rotations for <bilinear cancellation for badly approximable phases>.
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