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

Abstract: 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 $\delta(E)$.