Title: Borel-Bernstein theorem and Hausdorff dimension of sets in power-2-decaying Gauss-like expansion Show Chinese title

Title: 波莱尔-伯恩斯坦定理和幂2衰减高斯类似展开中集合的豪斯多夫维数

Abstract: Each $x\in (0,1]$ can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of $$ x=\sum_{i=1}^{\infty}2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))},\qquad d_i(x)\in \mathbb{N}. $$ Let $\phi:\mathbb{N}\to \mathbb{R}^{+}$ be an arbitrary positive function. We are interested in the size of the set $$F(\phi)=\{x\in (0,1]:d_n(x)\ge \phi(n)~~\text{i.m.}~n\}.$$ We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of $F(\phi)$. We also obtain the Hausdorff dimension of $F(\phi)$. Show Chinese abstract

Abstract: 每个$x\in (0,1]$都可以唯一地展开为一个幂-2衰减的高斯类似展开，形式为$$ x=\sum_{i=1}^{\infty}2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))},\qquad d_i(x)\in \mathbb{N}. $$。令$\phi:\mathbb{N}\to \mathbb{R}^{+}$为一个任意的正函数。我们感兴趣的是集合$$F(\phi)=\{x\in (0,1]:d_n(x)\ge \phi(n)~~\text{i.m.}~n\}.$$的大小。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。我们还得到了$F(\phi)$的豪斯多夫维数。