Group action and $L^2$-norm estimates of geometric problems

TL;DR

This paper applies a group theoretic framework to derive new results on the distribution of geometric configurations, focusing on distance sets, direction sets, and difference set scales, extending prior work from 2017.

Contribution

It introduces novel applications of the group theoretic approach to analyze product and quotient of distance sets and $L^2$-norms in geometric problems.

Findings

Results on product and quotient of distance sets

Estimates of $L^2$-norm of direction sets

Bounds on $L^2$-norm of scales in difference sets

Abstract

In 2017, by using the group theoretic approach, Bennett, Hart, Iosevich, Pakianathan, and Rudnev obtained a number of results on the distribution of simplices and sum-product type problems. The main purpose of this paper is to give a series of new applications of their powerful framework, namely, we focus on the product and quotient of distance sets, the $L^2$-norm of the direction set, and the $L^2$-norm of scales in difference sets.