论文标题
固定的Mattila-Sjölin类型定理用于产品集
A pinned Mattila-Sjölin type theorem for product sets
论文作者
论文摘要
我们概括了麦当劳和泰勒的结果，该结果涉及集合$ C_1 \ times C_2 $的边缘长度的大小，利用厚度的概念。具体来说，我们表明$ C_1，C_2 \ subset \ Mathbb {r}^d $ compact设置具有厚度满足$τ（C_1）τ（C_1）τ（C_2）> 1 $，然后以$ C_1 \ times C_2 $与任何固定的固定有限的有限型树构型配置相通用的边缘长度。我们最初是由麦当劳和泰勒（McDonald）和泰勒（Taylor）在实际线路上的康托尔（Cantor）设置的，我们使用了Falconer和Yavicoli引入的厚度概念，这使我们能够将麦当劳和泰勒的结果推广到$ \ mathbb {r}^d $中。
We generalize a result of McDonald and Taylor which concerns the size of the tuples of edge lengths in the set $C_1 \times C_2$ utilizing the notion of thickness. Specifically, we show that $C_1, C_2 \subset \mathbb{R}^d$ compact sets with thickness satisfying $τ(C_1) τ(C_2) >1$, then the edge lengths in $C_1 \times C_2$ corresponding to any pinned finite tree configuration has non-empty interior. Originally proven for Cantor sets on the real line by McDonald and Taylor, we use the notion of thickness introduced by Falconer and Yavicoli which allows us to generalize the result of McDonald and Taylor to compact sets in $\mathbb{R}^d$.
