关于某些正规数集的可数交的豪斯多夫维数
On the Hausdorff dimension of countable intersections of certain sets of normal numbers
摘要：
我们证明了$Q$ -分布正态但不是简单的$Q$ -比率正态的数的集合具有完全的豪斯多夫维数。进一步证明了在一定条件下，这种形式的集合的可数交即使不是胜利集（在W. Schmidt）。因此，我们构造了许多明确的例子，这些数字是同时分布正常的，但不是简单的比例正常的某些可数家庭的基本序列。此外，我们还证明了某些相关集是获胜集或第一范畴集。
英文摘要：
We show that the set of numbers that are $Q$-distribution normal but not simply $Q$-ratio normal has full Hausdorff dimension. It is further shown under some conditions that countable intersections of sets of this form still have full Hausdorff dimension even though they are not winning sets (in the sense of W. Schmidt). As a consequence of this, we construct many explicit examples of numbers that are simultaneously distribution normal but not simply ratio normal with respect to certain countable families of basic sequences. Additionally, we prove that some related sets are either winning sets or sets of the first category.
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