Effective equidistribution of polynomial trajectories in SL₃(ℝ)/SL₃(ℤ)

@misc{xing2026effective,
  title = {{Effective equidistribution of polynomial trajectories in $\operatornameSL_3(\mathbb R)/\operatornameSL_3(\mathbb Z)$}},
  author = {Hao Xing},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20513},
  eprint = {2610.20513},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20513},
}

TY  - UNPB
TI  - Effective equidistribution of polynomial trajectories in SL_3(ℝ)/SL_3(ℤ)
AU  - Hao Xing
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20513
UR  - https://arxiv.org/abs/2610.20513
AB  - For a fixed matrix-polynomial trajectory in SL_3(ℝ)/SL_3(ℤ), we give a necessary and sufficient intrinsic Diophantine condition for its time averages to equidistribute at a power rate on its homogeneous orbit closure. The condition uses a lattice in the space of polynomial functions on the normalized orbit-closure group, each admitting a representative of total degree at most two in the matrix entries. These polynomial tests have a common degree bound of two, independent of the curve degree and optimal in general. The proof begins with a classification of the possible orbit-closure groups and derives lower bounds for polynomial coefficient vectors from the intrinsic condition. It combines quantitative nilmanifold equidistribution, unipotent tangent approximation, and effective flow estimates, with arithmetic separation and quantitative nondivergence controlling cusp excursions and periodic-orbit obstructions.
ER  -