The density of sets containing large similar copies of finite sets

TL;DR

This paper establishes a density threshold for Lebesgue-measurable sets in Euclidean space to contain similar copies of all large finite point sets, with the threshold approaching 1 as the set size increases.

Contribution

It proves a density condition ensuring the presence of all large similar copies of finite sets and constructs examples showing the threshold's asymptotic behavior.

Findings

Sets with density > (n-2)/(n-1) contain all large similar copies of any n-point set.

The required density approaches 1 at a rate of 1 - O(n^{-1/5} log n).

The uniformity of scale for containing these copies is established.

Abstract

We prove that if $E \subseteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-measurable set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point set $P$ at all sufficiently large scales. Moreover, `sufficiently large' can be taken to be uniform over all $P$ with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of $n$-point sets tends to $1$ at a rate $1- O(n^{-1/5}\log n)$.