Distances in positive density sets in Rd

AbstractWe show that for a subset A of Rd with positive upper density, there is an R>0 such that for any r>R, there exist x and y in A with d(x,y)=r. The proof is based on the well-known second moment method in probability

The density of sets containing large similar copies of finite sets

Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.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).PreprintPeer reviewe