Metrical theory of signed Engel expansions

Motivated by the Engel and Pierce expansions, we introduce a signed Engel expansion. We expand each $x\in (0,1)\setminus \mathbb{Q}$ uniquely as $$x=\frac{{\epsilon }_1(x)}{d_1(x)}+\frac{{\epsilon }_2(x)}{d_1(x)d_2(x)}+\cdots +\frac{{\epsilon }_n(x)}{d_1(x)d_2(x)\cdots d_n(x)}+\cdots ,$$ where ${\epsilon }_1(x)1$ and ${\epsilon }_n(x)\in \{1,-1\}$ for $n\ge 2$. The digit sequence ${\{d_n(x)\}}_{n\ge 1}$ satisfying $d_{n+1}(x)\ge d_n(x)+2$ when ${\epsilon }_{n+1}(x)=-{\epsilon }_n(x)$ forms a non-decreasing sequence of even positive integers tending to infinity. On the one hand, we obtain the law of large numbers, the central limit theorem and the law of the iterated logarithm regarding $d_n(x)$ and ${\mathrm{\Delta }}_n(x)d_n(x)-d_{n-1}(x)(n\ge 2)({\mathrm{\Delta }}_1(x)d_1(x))$. On the other hand, we prove a Borel–Bernstein theorem on the zero-one law on the Lebesgue measure of the set $$\{x\in (0,1)\text{\hspace{0.17em}⁣}:R_n(x)\ge \phi (n)\text{ for infinity many }n\},$$ where $R_n(x)\frac{d_n(x)}{d_{n-1}(x)}(n\ge 2)(R_1(x)d_1(x))$ and $\phi$ is an arbitrary positive function defined on the set of positive integers.