We prove effective density theorems, with a polynomial error rate, for orbits of the upper triangular subgroup $${\mathrm{SL}}_2({\mathbb {R}})$$ in arithmetic quotients {C}})$$ and {R}})\times {\mathrm{SL}}_2({\mathbb . The proof is based on use Margulis function, tools from incidence geometry, spectral gap ambient space.