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We prove effective density theorems, with a polynomial error rate, for orbitsof the upper triangular subgroup of $\operatorname{SL}_2(\mathbb R)$ inarithmetic 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 incidencegeometry, and the spectral gap of the ambient space.

Polynomial effective density in quotients of $\mathbb H^3$ and $\mathbb H^2\times\mathbb H^2$