Affine Random Walks on the Torus

Abstract We study quantitative equidistribution of random walks on the torus by affine transformations. Under assumption that Zariski closure group generated linear part acts strongly irreducibly ${{\mathbb{R}}}^d$ and is either connected or contains a proximal element, we give estimates (depending only walk) for how fast walk equidistributes unless initial point translation transformations can be perturbed so trapped in finite orbit small cardinality. In particular, prove law to Haar measure if not orbit.