2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61009Using 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.7 pages; to appear in DIMACS volume "Unusual Applications of Number Theory"; related work at http://www.math.hmc.edu/~su/papers.htmlProbabilityNumber Theory60B15 (Primary) 11J13, 11K38, 11K60 (Secondary)text