  Our paper gives bounds for the rate of convergence for a class of random
walks on the d-dimensional torus generated by a set of n vectors in R^d/Z^d. We
give bounds on the discrepancy distance from Haar measure; our lower bound
holds for all such walks, and if the generators arise from the rows of a "badly
approximable" matrix, then there is a corresponding upper bound. The bounds are
sharp for walks on the circle.
