Generating an equidistributed net on a unit n-sphere using random rotations

We develop a randomized algorithm (that succeeds with high probability) for generating an $ϵ$-net in a sphere of dimension n. The basic scheme is to pick $O(n\ln (1\mathrm{/}n)+\ln (1\mathrm{/}\delta ))$ random rotations and take all possible words of length $O(n\ln (1\mathrm{/}ϵ))$ in the same alphabet and act them on a fixed point. We show this set of points is equidistributed at a scale of $ϵ$. Our main application is to approximate integration of Lipschitz functions over an n-sphere.