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is obtained for the rate of convergence with respect to (w. r. t.) the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N) w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence wrt the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no. 1, 23–45.] is sharp.