The density of sets containing large similar copies of finite sets

Abstract

We prove that if $E\subseteq {\mathbb{R}}^d$ ($d\ge 2$) 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 $1-O(n^{-1/5}\log n)$.

19 pages

References in corpus (4)

• A Szemerédi-type theorem for subsets of the unit cube
• Existence of similar point configurations in thin subsets of ${\mathbb{R}}^d$
• Boxes, extended boxes, and sets of positive upper density in the Euclidean space
• An elementary approach to simplexes in thin subsets of Euclidean space

Cited by in corpus (1)

• Large dilates of hypercube graphs in the plane