标题： 固定模式和$\mathbb R^d$中的密度定理 显示英文标题

标题： Pinned patterns and density theorems in $\mathbb R^d$

摘要： 对于整数$k\geq 3,d\geq 2,$，我们考虑在具有正上密度$\delta(E)$的$E\subseteq \mathbb R^d$中出现的固定点$k$-点模式的丰富性属性。 我们证明，对于任何固定的$k$点模式$V$，存在一个具有正上密度的集合$E$，使得$E$避免所有足够大的仿射副本$V$，其中有一个顶点固定在$E$中的任意点。 然而，我们得到一个定量结果，该结果指出，对于任何具有正上密度的固定$E$，存在一个$k$-点模式$V,$，使得对于任何$x\in E$，钉住缩放因子集\begin{equation*} D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\}, \end{equation*}具有上密度$\geq \tilde \varepsilon>0$，其中常数$\tilde \varepsilon$依赖于$k,d$和$\delta(E)$。 显示英文摘要

摘要： For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $\delta(E)$. We show that for any fixed $k$-point pattern $V$, there is a set $E$ with positive upper density such that $E$ avoids all sufficiently large affine copies of $V$, with one vertex fixed at any point in $E$. However, we obtain a positive quantitative result, which states that for any fixed $E$ with positive upper density, there exists a $k$-point pattern $V,$ such that for any $x\in E$, the pinned scaling factor set \begin{equation*} D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\}, \end{equation*} has upper density $\geq \tilde \varepsilon>0$, where constant $\tilde \varepsilon$ depends on $k,d$ and $\delta(E)$.