Effective equidistribution of unipotent orbits in homogeneous spaces of $\SL(2,\R)\ltimes(\R^2)^{k}$

TL;DR

This paper establishes effective equidistribution results for unipotent orbits in certain homogeneous spaces, using the circle method to achieve polynomial rates of convergence.

Contribution

It provides the first polynomially effective equidistribution results for unipotent orbits in these specific homogeneous spaces, employing the delta symbol circle method.

Findings

Polynomial effective asymptotic equidistribution for expanding translates of unipotent orbits.

Effective long-term equidistribution for individual unipotent orbits.

Application of the delta symbol circle method in this context.

Abstract

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $\Gamma$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially effective asymptotic equidistribution results for expanding translates of $u_{\R}$-orbits and for long pieces of individual $u_{\R}$-orbits in $\Gamma\backslash G$. An important ingredient of the proof is the delta symbol version of the circle method.