Badly approximable number (${\inf }_{q\ge 1}q\Vert q\alpha \Vert >0$)

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.

For the nearest integer $a$ to $q\sqrt{2}$, the nonzero integer $2q^2-a^2$ has absolute value at least one. Dividing by $q\sqrt{2}+a\le 4q$ proves the bound. Thus the points $0,\sqrt{2},\dots ,W\sqrt{2}$ modulo one are separated by at least $1/(4W)$. Ordering their distances from zero gives ${\sum }_{h\le W}\Vert h\sqrt{2}{\Vert }^{-1}\ll W\log (2W)$.