Points on SO(3) with low logarithmic energy

We describe several randomized collections of 3× 3 3 × 3 rotation matrices and analyze their associated logarithmic energy. The best one (that is, the one attaining the lowest expected logarithmic energy) is constructed by choosing r points on the sphere, which come from the zeros of a randomly chosen degree r polynomial, and considering at each of these points a set of s evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of $$n=rs$$ n = r s rotation matrices.