论文标题
$ \ mathbb {z}^{2} \ subset \ mathbb {r}^{2} $中的marstand类型切片语句
Marstrand type slicing statements in $\mathbb{Z}^{2}\subset \mathbb{R}^{2}$ are false for the counting dimension
论文作者
论文摘要
我们表明，对于$ \ r^{2} $的$ 1 $分离子集，天然的marstrand型切片语句是错误的，其中莫雷拉（Moreira）和利马（Moreira）和利马（Moreira）和利马（Moreira and Lima）和其中的变体在不同的上下文中介绍了。我们构建了该飞机的$ 1 $分离子集$ e $，该$ e $具有计数尺寸$ 1 $，而对于宽度$ 1 $的lebesgue测量量参数，该管与设定$ e $的交汇处具有计数尺寸$ 1 $。这与这种集合的行为与切片定理成立的质量维度相反。
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.
