The density of sets containing large similar copies of finite sets

Abstract

We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than n−2n−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−1/5log n).