Group action and L2-norm estimates of geometric problems

Abstract

In 2017, by using the group theoretic approach and tools from discrete Fourier analysis, 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 L2-norm of the direction set, and the L2-norm of scales in difference sets.