Metrical theory of signed Engel expansions

Abstract

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 $\epsilon_{1}(x)\coloneqq1$ and $\epsilon_{n}(x)\in\left\{1,-1\right\}$ for $n\geq2$. The digit sequence $\left\{d_{n}(x)\right\}_{n\geq1}$ satisfying $d_{n+1}(x)\geq d_{n}(x)+2$ 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 $\Delta_{n}(x)\coloneqq d_{n}(x)-d_{n-1}(x)\ (n\geq2)\ (\Delta_{1}(x)\coloneqq d_{1}(x))$. On the other hand, we prove a Borel--Bernstein theorem on the zero-one law on the Lebesgue measure of the set $$\left\{x\in(0,1)\colon R_{n}(x)\geq\phi(n)\ \textnormal{ for infinity many } n\right\},$$ where $R_{n}(x)\coloneqq\frac{d_{n}(x)}{d_{n-1}(x)}\ (n\geq2)\ (R_{1}(x)\coloneqq d_{1}(x))$ and $\phi$ is an arbitrary positive function defined on the set of positive integers.