Metrical theory of power-2-decaying Gauss-like expansion

Each $x\in (0,1]$ can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of \begin{equation*} x=\sum_{i=1}^{\infty}2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))},\qquad d_i(x)\in \mathbb{N}. \end{equation*} Let $\phi:\mathbb{N}\to \mathbb{R}^{+}$ be an arbitrary positive function. We are interested in the size of the set $$F(\phi)=\{x\in (0,1]:d_n(x)\ge \phi(n)~~\text{for infinity many}~n\}.$$ We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of $F(\phi)$. When the Lebesgue measure of $F(\phi)$ is zero, we calculate its Hausdorff dimension. Furthermore, we analyse the growth rate of the maximal digit among the first $n$ digits from probability and multifractal perspectives.