$\mathbb{Z}^{2}\subset \mathbb{R}^{2}$中Marstrand类型切片陈述在计数维度下是错误的
Marstrand type slicing statements in $\mathbb{Z}^{2}\subset \mathbb{R}^{2}$ are false for the counting dimension
AI总结：
研究指出，在$\mathbb{R}^{2}$中，计数维度下Marstrand类型切片陈述不成立，通过构造具有计数维度1的集合E，展示其与质量维度下的切片定理行为不同。
摘要：
我们证明，对于$\R^{2}$中的1分离子集，自然的Marstrand类型切片陈述在计数维度下不成立，该维度此前由Moreira和Lima使用过，其变体在不同上下文中也曾被引入。我们构造了一个$\mathbb{R}^{2}$中的1分离子集$E$，其具有计数维度1，而对于正勒贝格测度参数集中的宽度为1的管，管与集合$E$的交集具有计数维度1。这与质量维度下此类集合的行为形成对比，因为在质量维度下切片定理成立。
英文摘要：
We show that for $1$ separated subsets of $\R^{2}$, the natural Marstrand type slicing statements are false with the counting dimension that was used earlier by Moreira and Lima and variants of which were introduced earlier in different contexts. We construct a $1$ separated subset $E$ of the plane which has counting dimension $1$, while for a positive Lebesgue measure parameter set of tubes of width $1$, the intersection of the tube with the set $E$ has counting dimension $1$. This is in contrast to the behavior of such sets with the mass dimension where the slicing theorems hold true.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
