典型数与正规数之间差集的豪斯多夫维数
On the Hausdorff dimension of the difference sets between typical and normal numbers
AI总结：
该研究探讨典型数与基2正规数差集的豪斯多夫维数，证明二者差集具完全豪斯多夫维数，通过计数论证而非显式构造确定维数。
AI中文摘要：
我们研究典型数集与基2正规数集之间差集的豪斯多夫维数，其中典型数由控制数字最长游程的Erdős–Rényi定律定义。尽管这两种性质对勒贝格测度下几乎所有实数都成立，但二者互不蕴含。所得的差集TpcN∖Normal与Normal∖TpcN在博雷尔分层中为D₂(Π₃⁰)-完全集。我们证明这种逻辑复杂性在几何上也匹配：两个差集均具有完全豪斯多夫维数。与控制单个数字频率的经典Besicovitch型集不同，正规性要求所有有限块的频率同时正确，而适用于简单正规性的显式Moran集构造在该要求下失效。我们转而通过计数论证确定维数，无需显式构造。
英文摘要：
We study the Hausdorff dimension of the difference sets between the set of typical numbers, defined via the Erdős--Rényi law governing the longest run of digits, and the set of numbers normal in base $2$. Although both properties hold for Lebesgue-almost every real number, neither implies the other. The resulting difference sets $\textit{TpcN}\setminus\textit{Normal}$ and $\textit{Normal}\setminus\textit{TpcN}$ are $D_2({\bf Π}_3^0)$-complete in the Borel hierarchy. We show that this logical complexity is matched geometrically: both difference sets have full Hausdorff dimension. Unlike the classical Besicovitch-type sets governing single-digit frequencies, normality requires the correct frequency of every finite block simultaneously, and the explicit Moran-set constructions that succeed for simple normality break down under this requirement. We instead establish the dimension via a counting argument bypassing the need for an explicit construction.
