On the rate of convergence for $(\log_b n)$

Chuang Xu

Published 2016-09-26Version 1

In this paper, we study rate of convergence for the distribution of sequence of logarithms $(\log_bn)$ for integer base $b\ge2.$ It is well-known that the slowly growing sequence $(\log_bn)$ is not uniformly distributed modulo one. Its distributions converge to a loop of translated exponential distributions with constant $\log b$ in the space of probabilities on the circle. We give an upper bound for the rate of convergence under Kantorovich distance $d_{\mathbb{T}}$ on the circle. We also give a sharp rate of convergence under Kantorovich distance $d_{\mathbb{R}}$ on the line $\mathbb{R}.$ It turns out that the convergence under $d_{\mathbb{T}}$ is much faster than that under $d_{\mathbb{R}}.$