论文标题
在Furstenberg套装的包装上
On the packing dimension of Furstenberg sets
论文作者
论文摘要
我们证明，如果$α\在（0,1/2] $中，则一组$ e \ subset \ subset \ mathbb {r}^2 $的包装维度，其中存在一组尺寸$ 1 $的行$ e $ in Dimension $ \geα$中的$ 1 $至少是$ 1/2+α+c（live）$ c（live）$α+c（α）的$ c（某些$ c（某些）$ c（ $α$ -FURSTENBERG，即具有Hausdorff dimension $ \geα$与T. Orponen的较早结果的相交，这为$ -Furstenberg的包装维度提供了改进。线条，并表明$α$ -FURSTENBERG集合的尺度类似于一组接近$ 1/2+α$的尺寸，如果存在，则相当稀疏。
We prove that if $α\in (0,1/2]$, then the packing dimension of a set $E\subset\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\ge α$ is at least $1/2+α+c(α)$ for some $c(α)>0$. In particular, this holds for $α$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $\geα$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $α$-Furstenberg sets over the "trivial" estimate for all values of $α\in (0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $α$-Furstenberg set resembles a set of dimension close to $1/2+α$, if they exist, are rather sparse.
