2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68427Our 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.10 pages; related work at http://www.math.hmc.edu/~su/papers.htmlProbability60B15; 11J13, 11K38text