Contains an analysis of the random walk on the circle generated by an irrational rotation, using discrepancy to measure convergence to the uniform distribution. Sharp rates are obtained for quadratic irrationals (order k^(-1/2)), somewhat less sharp results for all other irrationals based on their "type" classification.   A key ingredient in the analysis is an interesting recurrence relation for the distribution of irrational multiples (mod 1). This is used to bound terms involving exponential sums.