Title: Metrical theory of power-2-decaying Gauss-like expansion Show Chinese title

Title: 幂次为2衰减的类似高斯展开的度量理论

Abstract: 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. Show Chinese abstract

Abstract: 每个 $x\in (0,1]$ 可以唯一地展开为一个幂-2衰减的高斯型展开，形式为 \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*} 设 $\phi:\mathbb{N}\to \mathbb{R}^{+}$ 为一个任意的正函数。 我们感兴趣的是集合的大小 $$F(\phi)=\{x\in (0,1]:d_n(x)\ge \phi(n)~~\text{for infinity many}~n\}.$$我们证明了一个关于 $F(\phi)$的勒贝格测度的零一律的波莱尔-伯恩斯坦定理。 当 $F(\phi)$的勒贝格测度为零时，我们计算其 豪斯多夫维数。 此外，我们从概率和多重分形的角度分析了前 $n$位中最大数字的增长率。