Effective Ratner equidistribution result for SL$(2,\mathbb R)\ltimes \mathbb R^{2k}$ and applications to quadratic forms

Let $G=\mathrm{SL}(2,\mathbb R)\ltimes \mathbb R^{2k}$ and let $\Gamma$ be a congruence subgroup of $\mathrm{SL}(2,\mathbb Z)\ltimes\mathbb Z^{2k}$. We give an effective equidistribution result for a family of 1-dimensional unipotent orbits in $\Gamma\backslash G$. The proof involves Spectral methods and bounds for exponential sums. We apply this result to obtain an effective Oppenheim type result for a class of indefinite irrational quadratic forms. This is based on a joint work with Andreas Strombergsson.

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