Pinned patterns and density theorems in ${\mathbb{R}}^d$

For integers $k\ge 3,d\ge 2,$ 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 $\ge \tilde{\epsilon }>0$, where constant $\tilde{\epsilon }$ depends on $k,d$ and $\delta (E)$.