The positive finite-rate conjecture for the distributional asymptotics of $(\log_b n)$ modulo 1

Let $b$ be the base, let $\nu_N$ denote the empirical distribution associated with the fractional parts of $(\log_b n)$ up to $N$, let $E_b$ be the relevant exponential distribution, let $R^{-1}_{\langle-\log_b N\rangle}$ denote rotation by $-\log_b N$ modulo $1$, and let $d_{\mathbb{T}}$ be the metric on probability measures on the circle $\mathbb{T}$. Positive finite-rate conjecture.

$$0<\liminf_{N\to\infty}\frac{N}{\sqrt{\log N}}d_{\mathbb{T}}(\nu_N,E_b\circ R^{-1}_{\langle-\log_bN\rangle})\le\limsup_{N\to\infty}\frac{N}{\sqrt{\log N}}d_{\mathbb{T}}(\nu_N,E_b\circ R^{-1}_{\langle-\log_bN\rangle})<\infty.$$

The conjecture asserts that the normalized convergence rate is bounded above and bounded away from zero, complementing the paper's upper-bound estimate and predicting the correct order of convergence.