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

We prove an effective density theorem with polynomial error rate for orbits of upper triangular subgroup of ${\mathrm{S}\mathrm{L}}_2({\mathbb{Q}}_p)$ in ${\mathrm{S}\mathrm{L}}_2({\mathbb{Q}}_p)\times {\mathrm{S}\mathrm{L}}_2({\mathbb{Q}}_p)$ 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.