Random circle rotations

If at least one of $\alpha$ or $\beta$ is an irrational multiple of $\pi$ then you do get equidistribution (almost surely). I'm sure this is a folklore type result; but one place where I know it can be found is a paper of Lagarias and Soundararajan (http://arxiv.org/pdf/math/0509175.pdf) which appeared in JLMS. See Theorem 4.1 there. The context for that paper is that the $(3x+1)$ map where one multiplies either by a number close to $3/2$ or multiplies by $1/2$ at least initially follows such a pattern, and therefore exhibits Benford's law.