Second moment of the light-cone Siegel transform and applications

We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form $Q$ to a function on the homogenous space ${\text{SO}}_Q^{+}(\mathbb{Z})\mathrm{\backslash }{\text{SO}}_Q^{+}(\mathbb{R})$. In particular, we prove a second moment formula for this transform for forms of signature $(n+1,1)$, and show how it can be used for various applications for counting integer points on the light cone. In particular, we prove some new results on intrinsic Diophantine approximations on ellipsoids as well as on the distribution of values of random linear and quadratic forms on the light cone.