论文标题
低维度的Mattila-Sjölin定理的简单定理
A Mattila-Sjölin theorem for simplices in low dimensions
论文作者
论文摘要
在本文中，我们表明，如果一个紧凑的集合$ e \ subset \ mathbb {r}^d $，$ d \ geq 3 $，则具有大于$ \ frac {（4K-1）} {4K} {4K} d+\ frac {1}} $ 3 \ 3 \ leq时 d<\frac{k(k+3)}{(k-1)}$ or $d- \frac{1}{k-1}$ when $\frac{k(k+3)}{(k-1)} \leq d$, then the set of congruence class of simplices with vertices in $E$ has nonempty interior.通过$ e $中的顶点的一组简单类别的一组，我们的意思是$δ_{k}（e）= \ left \ {\ vec {\ vec {t} =（t_ {ij}）：| x_i-x_j | = t_ | = t_j {ij {ij {ij}; \ x_i，x_j \ in E; \ 0 \ leq i <j \ leq k \ right \} \ subset \ mathbb {r}^{\ frac {k（k+1）} {2}} $ $，其中$ 2 \ leq leq k <d $。这一结果改善了我们以前的工作，从某种意义上说，我们现在可以获得Hausdorff尺寸阈值，这使我们能够确保当$ e $的三倍的分三角形形成的一整套三角形类别时，当$ D = 3 $以及扩展到所有简单时。可以将目前的工作视为Mattila-Sjölin定理的扩展，该定理为距离集建立了非空内饰，而不是简单的一致类别。
In this paper we show that if a compact set $E \subset \mathbb{R}^d$, $d \geq 3$, has Hausdorff dimension greater than $\frac{(4k-1)}{4k}d+\frac{1}{4}$ when $3 \leq d<\frac{k(k+3)}{(k-1)}$ or $d- \frac{1}{k-1}$ when $\frac{k(k+3)}{(k-1)} \leq d$, then the set of congruence class of simplices with vertices in $E$ has nonempty interior. By set of congruence class of simplices with vertices in $E$ we mean $$Δ_{k}(E) = \left \{ \vec{t} = (t_{ij}) : |x_i-x_j|=t_{ij} ; \ x_i,x_j \in E ; \ 0\leq i < j \leq k \right \} \subset \mathbb{R}^{\frac{k(k+1)}{2}}$$ where $2 \leq k <d$. This result improves our previous work in the sense that we now can obtain a Hausdorff dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of $E$ has nonempty interior when $d=3$ as well as extending to all simplices. The present work can be thought of as an extension of the Mattila-Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.
