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