Cram\'er-Type Moderate Deviations for Engel's Series via a Martingale Approach

Let $x$ be uniformly distributed on $(0,1)$, and let $(q_n)_{n\geq1}$ be the digits of its Engel series expansion. We establish a Cram\'er-type moderate deviation expansion for $(\log q_n-n)/\sqrt n$. The proof is based on a martingale decomposition and asymptotic results for martingales. As consequences, we obtain a moderate deviation principle over the full range of scales between the central limit theorem and the law of large numbers, without the additional lower rate restriction required in several earlier works. We also derive a uniform Berry--Esseen bound of order $(\log n)/\sqrt n$.