  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.
