We prove effective density theorems, with a polynomial error rate, for orbits of the upper triangular subgroup of SL 2(R) in arithmetic quotients of SL 2(C) and SL 2(R) × SL 2(R). The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.
Bibliographical noteFunding Information:
E.L. acknowledges support by ERC 2020 Grant HomDyn (Grant No. 833423). A.M. acknowledges support by the NSF Grants DMS-1764246 and 2055122.
We would like to thank the Hausdorff Institute for its hospitality during the winter of 2020. A.M. would like to thank the Institute for Advanced Study for its hospitality during the fall of 2019 where parts of this project were carried out. The authors would like to thank Gregory Margulis and Nimish Shah for many discussions about effective density, and Joshua Zahl for helpful communications regarding projections theorems. We would also like to thank Zhiren Wang with whom we discussed related questions. We thank the anonymous referees for their helpful comments.
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.