Low dimensional pinned distance sets via spherical averages

TL;DR

This paper introduces a new inequality for the average t-energy of pinned distance measures, refining existing theorems and extending results to lower-dimensional sets in geometric measure theory.

Contribution

It derives a novel inequality for pinned distance measures, extending Mattila's theorem and providing an analogue of Liu's theorem for sets with dimension less than one.

Findings

Refined inequality for average t-energy of pinned distance measures

Extension of Mattila's theorem to pinned distance sets

Analogue of Liu's theorem for low-dimensional sets

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.