乘积集的一个Pinned Mattila-Sjölin型定理
A pinned Mattila-Sjölin type theorem for product sets
摘要：
我们推广了McDonald和Taylor的结果，该结果涉及利用厚度概念的集合$C_1 \times C_2$中边长元组的大小。具体地说，我们证明了厚度满足$τ(C_1) τ(C_2) >1$的$C_1, C_2 \subset \mathbb{R}^d$紧集，则对应于任意固定有限树配置的$C_1 \times C_2$中的边长具有非空内部.最初证明康托集上的真正的线麦当劳和泰勒，我们使用的概念，厚度介绍了法尔科纳和Yavicoli，使我们能够推广的结果麦当劳和泰勒紧集$\mathbb{R}^d$。
英文摘要：
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 $\mathbb{R}^d$.
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