Low dimensional pinned distance sets via spherical averages

Abstract

An inequality is derived for the average $t$-energy of pinned distance measures, where $0<t<1$. This refines Mattila's theorem on distance sets to pinned distance sets, and gives an analogue of Liu's theorem for pinned distance sets of dimension smaller than 1.

5 pages. Accepted version

References in corpus (2)

• An $L^2$-identity and pinned distance problem
• An improved result for Falconer's distance set problem in even dimensions