A useful lemma for calculating the Hausdorff dimension of certain sets in Engel expansions

Abstract

Let $\{s_n\}$ and $\{t_n\}$ be two sequences of positive real numbers. Under some mild conditions on $\{s_n\}$ and $\{t_n\}$, we give the precise formula of the Hausdorff dimension of the set \[ \mathbb{E}(\{s_n\},\{t_n\}):=\Big\{x\in(0,1): s_{n}<d_{n}(x)\leq s_n+t_n, \forall n\geq1\Big\}, \] where $d_n(x)$ denotes the digit of the Engel expansion of $x$. This result improves the Lemma 2.6 of Shang and Wu (2021JNT), and is very useful for calculating the Hausdorff dimension of certain sets in Engel expansions.