论文标题
$ \ mathbb {r}^n $中的行联盟
Unions of lines in $\mathbb{R}^n$
论文作者
论文摘要
我们证明了D. Oberlin在$ \ mathbb {r}^n $中的工会的维度上的猜想。如果$ d \ geq 1 $是一个整数，$ 0 \ leqβ\ leq 1 $，而$ l $是$ \ mathbb {r}^n $中的一组行，带有hausdorff dimension至少$ 2（d-1） +β$，那么$ l $ l $ l $ l $ d $ d +β$ d +β。我们的证明结合了Carbery和Valdimarsson的多线性Kakeya定理的精致版本，以及Bourgain和Guth的多线性与线性论证。
We prove a conjecture of D. Oberlin on the dimension of unions of lines in $\mathbb{R}^n$. If $d \geq 1$ is an integer, $0 \leq β\leq 1$, and $L$ is a set of lines in $\mathbb{R}^n$ with Hausdorff dimension at least $2(d-1) + β$, then the union of the lines in $L$ has Hausdorff dimension at least $d + β$. Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear to linear argument of Bourgain and Guth.
