Effective density for inhomogeneous quadratic forms I: generic forms and fixed shifts

TL;DR

This paper proves effective density results for inhomogeneous quadratic forms with fixed or generic shifts, extending Oppenheim's conjecture and providing optimal density bounds under various conditions.

Contribution

It establishes effective versions of Oppenheim's conjecture for inhomogeneous forms with fixed and generic shifts, introducing new technical tools involving Siegel transforms.

Findings

Optimal density for values of generic inhomogeneous forms with rational shifts

Density results for fixed irrational shifts satisfying Diophantine conditions

A new formula for the second moment of Siegel transforms on congruence quotients

Abstract

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the second moment of Siegel transforms on certain congruence quotients of $\operatorname{SL}_n(\mathbb{R})$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.