 Badly approximable number

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