  Using the discrepancy metric, we analyze the rate of convergence of a random
walk on the circle generated by d rotations, and establish sharp rates that
show that badly approximable d-tuples in R^d give rise to walks with the
fastest convergence. We use the discrepancy metric because the walk does not
converge in total variation. For badly approximable d-tuples, the discrepancy
is bounded above and below by (constant)k^(-d/2), where k is the number of
steps in the random walk. We show how the constants depend on the d-tuple.
