Polynomial effective density in quotient of $\mathrm{SL}_2(\mathbb{Q}_p) \times \mathrm{SL}_2(\mathbb{Q}_p)$

TL;DR

This paper establishes a polynomial error rate effective density theorem for orbits of an upper triangular subgroup in the product of two $ ext{SL}_2(Q_p)$ groups, utilizing advanced harmonic analysis and spectral gap techniques.

Contribution

It provides the first polynomial error rate effective density result for these orbits in the $p$-adic setting, extending previous qualitative density results.

Findings

Proves polynomial error rate effective density theorem for $ ext{SL}_2(Q_p)$ orbits.

Uses Margulis function and projection theorem in $Q_p^3$.

Employs spectral gap to achieve quantitative bounds.

Abstract

We prove an effective density theorem with polynomial error rate for orbits of upper triangular subgroup of $\mathrm{SL}_2(\mathbb{Q}_p)$ in $\mathrm{SL}_2(\mathbb{Q}_p) \times \mathrm{SL}_2(\mathbb{Q}_p)$ for prime number $p > 3$. The proof is based on the use of Margulis function, a restricted projection theorem on $\mathbb{Q}_p^3$, and spectral gap of the ambient space.