Second moment of the light-cone Siegel transform and applications

TL;DR

This paper establishes a second moment formula for the light-cone Siegel transform related to quadratic forms of signature (n+1,1), enabling new results in counting integer points and Diophantine approximation on ellipsoids.

Contribution

It introduces a second moment formula for the light-cone Siegel transform and applies it to problems in counting integer points and Diophantine approximation on ellipsoids.

Findings

Derived a second moment formula for the light-cone Siegel transform.

Applied the formula to count integer points on light cones.

Obtained new results on Diophantine approximation on ellipsoids.

Abstract

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 $\text{SO}^+_Q(\mathbb{Z})\backslash \text{SO}^+_Q(\mathbb{R})$. 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.