AI 中文总结
该数学研究确定一维精确逼近集笛卡尔积的豪斯多夫与填充维数，改进均匀分布系统框架建立对应乘积定理，拓展了相关逼近集维数的研究结论。
AI 中文摘要
我们确定一维精确逼近集的笛卡尔积的豪斯多夫维数与填充维数。我们的主要结果建立了Wang和Wu（2024）近期关于上极限逼近集的乘积定理在精确逼近集下的对应结论，表明转向小得多的精确逼近集（下极限集）并不会减小笛卡尔积的豪斯多夫维数。关键要素之一是对Bandi-Ghosh-Nandi（2023）的均匀分布系统框架进行了改进，利用有理点的精细算术分布，进而在更弱的收敛条件下给出了该乘积集的豪斯多夫维数。
英文摘要
We determine the Hausdorff and packing dimensions of Cartesian products of one-dimensional exact approximation sets. Our main result establishes the exact-approximation counterpart of the recent product theorem of Wang and Wu (2024) for limsup approximation sets, showing that passing to the substantially smaller exact approximation sets (liminf sets) does not reduce the Hausdorff dimension of the Cartesian product. One of the key ingredients is a refinement of the well-distributed-system framework of Bandi--Ghosh--Nandi (2023) by exploiting the fine arithmetic distribution of rational points which then gives the Hausdorff dimension of the product set under a weaker convergence condition.
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