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

TL;DR

This paper investigates the occurrence and avoidance of scaled and translated fixed point patterns in dense subsets of Euclidean space, revealing both limitations and quantitative abundance results.

Contribution

It demonstrates the existence of dense sets avoiding large affine copies of fixed patterns and establishes a positive density result for the set of scaling factors at each point.

Findings

Existence of dense sets avoiding large affine copies of fixed patterns.

Positive lower density of scaling factors for patterns in any dense set.

Quantitative bounds depending on pattern, dimension, and density.

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 $\delta(E)$.