Central limit theorem for Benford errors

Let $a>0$ be a real number satisfying the relevant condition, and write $\alpha=\log_{10}a$. Suppose that the continued fraction expansion of $\alpha$ satisfies

$$\lim_{k\to\infty}\frac{a_k^2}{\sum_{i=1}^k a_i^2}=0.$$

For a digit $d\in\{1,2,\dots,9\}$, let $E_d(N,\{a^n\})$ denote the Benford error, and assume that $d$ does not satisfy the bounded-error condition. Central limit theorem for Benford errors. There exist sequences $\{A_N\}$ and $\{B_N\}$ such that, for all $u<v$,

$$\lim_{N\to\infty}\frac1N\#\left\{n\le N:u\le\frac{E_d(N,\{a^n\})-A_N}{B_N}<v\right\}=\frac{1}{\sqrt{2\pi}}\int_u^v e^{-x^2/2}\,dx.$$

This conjecture applies Beck's heuristic for interval discrepancies to Benford errors. It predicts Gaussian fluctuations whenever the continued-fraction coefficients satisfy the stated Lindeberg-type condition and the error is not bounded; the source does not provide a resolution.