A special case of effective Ratner equidistribution and applications to quadratic forms

Title: Effective Ratner equidistribution for $\text{SL}(2,\mathbb R)\ltimes(\mathbb R^2)^{\oplus k}$ and applications to quadratic forms

Abstract: Let $G=\text{SL}(2,\mathbb R)\ltimes(\mathbb R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of $\text{SL}(2,\mathbb Z)\ltimes(\mathbb Z^2)^{\oplus k}$. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the homogeneous space $\Gamma\backslash G$. The proof involves spectral analysis and use of Weil's bound on Kloosterman sums. I will also discuss applications to effective results for variants of the Oppenheim conjecture on the density of $Q(\mathbb Z^n)$ on the real line, where $Q$ is an irrational indefinite quadratic form.

(Based on joint work with Pankaj Vishe.)