On distinct distances in homogeneous sets in the Euclidean space

Abstract

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega (n^{2d/(d^2+1)}/{\log }^{c(d)}n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega (n^{.6091})$ distinct distances.

References in corpus (2)

• Some connections between Falconer's distance set conjecture, and sets of Furstenburg type
• Fourier bases and a distance problem of Erd\H os

Cited by in corpus (4)

• A Szemeredi-Trotter type theorem in ${\mathbb{R}}^4$
• On a question of Bourgain about geometric incidences
• An incidence theorem in higher dimensions
• On distance measures for well-distributed sets