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Density of almost squares of horospherical orbits in non-uniform quotients of SL2(ℝ)×SL2(ℝ)

  • Konstantin Andritsch
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Abstract

Let Γ⊂SL2(ℝ)×SL2(ℝ) be an irreducible, non-uniform lattice and define the space X = SL2(ℝ)×SL2(ℝ)/Γ. Let U be the standard horospherical subgroup in SL2(ℝ)×SL2(ℝ). We show that for every δ>0 and x∈ X with dense U-orbit, the U-orbit evaluated at almost squares, {(u(n2−δ,m2−δ) : n,m∈ℕ}· x, is dense in X. The main idea of the proof is to shadow large periodic U-orbits and reduce the statement to a density statement within the periodic U-orbit, i.e. a statement for the $2$-torus 𝕋2. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic U-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \cite{KR25} of \citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in SL2(ℝ).

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Cite

BibTeXRIS
BibTeX
@misc{andritsch2026density,
  title = {{Density of almost squares of horospherical orbits in non-uniform quotients of $\operatornameSL_2(\mathbbR)\times\operatornameSL_2(\mathbbR)$}},
  author = {Konstantin Andritsch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24752},
  eprint = {2609.24752},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24752},
}
RIS
TY  - UNPB
TI  - Density of almost squares of horospherical orbits in non-uniform quotients of SL_2(ℝ)×SL_2(ℝ)
AU  - Konstantin Andritsch
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24752
UR  - https://arxiv.org/abs/2609.24752
AB  - Let Γ⊂SL_2(ℝ)×SL_2(ℝ) be an irreducible, non-uniform lattice and define the space X = SL_2(ℝ)×SL_2(ℝ)/Γ. Let U be the standard horospherical subgroup in SL_2(ℝ)×SL_2(ℝ). We show that for every δ>0 and x∈ X with dense U-orbit, the U-orbit evaluated at almost squares, {(u_(n^2−δ,m^2−δ) : n,m∈ℕ}· x, is dense in X. The main idea of the proof is to shadow large periodic U-orbits and reduce the statement to a density statement within the periodic U-orbit, i.e. a statement for the $2$-torus 𝕋^2. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic U-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \cite{KR25} of \citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in SL_2(ℝ).
ER  -