Density of almost squares of horospherical orbits in non-uniform quotients of $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$

Abstract

Let $Γ\subset\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$ be an irreducible, non-uniform lattice and define the space $X = \operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})/Γ$. Let $U$ be the standard horospherical subgroup in $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$. We show that for every $δ>0$ and $x\in X$ with dense $U$-orbit, the $U$-orbit evaluated at almost squares, $\{(u_{(n^{2-δ},m^{2-δ})}~:~n,m\in\mathbb{N}\}\cdot x$, is dense in $X$. The main idea of the proof is to shadow large periodic $U$-orbits and reduce the statement to a density statement within the periodic $U$-orbit, i.e. a statement for the $2$-torus $\mathbb{T}^2$. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic $U$-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \cite{KR25} of \citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in $\operatorname{SL}_2(\mathbb{R})$.