K-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy
Abstract
Let q be a prime power, F=𝔽q( (t−1) ), G=SL2(F), Γ=SL2(𝔽q[t]), and K=SL2(𝔽q[ [t−1] ]), and let U<G be the upper unipotent subgroup. We study right K-spherical averages along U on X=Γbackslash G. Expanding translates of compact U-orbits and compact-open Funicodex00F8lner-ball averages become terminal layers of rooted descendant shadows in the Bruhat--Tits tree. In the even sector, we compute the Haar height law and signed finite-scale discrepancy exactly. This yields K-spherical equidistribution for compact-orbit translates and, for irrational boundary endpoints, for Funicodex00F8lner-ball averages. For a depth-N shadow rooted at height k, with cutoff M=N−k≥0, bounded-profile errors are Oq(q−M) in the backward state and Oq(q−M/2) uniformly for moving roots, while the shadow law eventually agrees exactly with the Haar law on every fixed finite height window. For |Φ(2m)|≤ Cqαm, α<2, three rate regimes arise, with a linear-in-scale factor at α=1 and explicit moving-root dependence. Artin continued-fraction digits eventually encode the cutoff and these rates excursion by excursion through individual digit degrees.
Related
- Spherical Averages on nilpotent Groups
- Sharp decay of the spherical averages of the Fourier transform of open sets
- A set of points with optimal L2 spherical cap discrepancy
- Equivariant compactifications of a unipotent group by a smooth projective horospherical variety of Picard number one
- Density of almost squares of horospherical orbits in non-uniform quotients of SL2(ℝ)×SL2(ℝ)
BibTeX
@misc{kwon2026kspherical,
title = {{$K$-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy}},
author = {Sanghoon Kwon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08704},
eprint = {2607.08704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - K-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy
AU - Sanghoon Kwon
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2607.08704
UR - https://arxiv.org/abs/2607.08704
AB - Let q be a prime power, F=𝔽_q( (t^−1) ), G=SL_2(F), Γ=SL_2(𝔽_q[t]), and K=SL_2(𝔽_q[ [t^−1] ]), and let U<G be the upper unipotent subgroup. We study right K-spherical averages along U on X=Γbackslash G. Expanding translates of compact U-orbits and compact-open Funicodex00F8lner-ball averages become terminal layers of rooted descendant shadows in the Bruhat--Tits tree. In the even sector, we compute the Haar height law and signed finite-scale discrepancy exactly. This yields K-spherical equidistribution for compact-orbit translates and, for irrational boundary endpoints, for Funicodex00F8lner-ball averages. For a depth-N shadow rooted at height k, with cutoff M=N−k≥0, bounded-profile errors are O_q(q^−M) in the backward state and O_q(q^−M/2) uniformly for moving roots, while the shadow law eventually agrees exactly with the Haar law on every fixed finite height window. For |Φ(2m)|≤ Cq^αm, α<2, three rate regimes arise, with a linear-in-scale factor at α=1 and explicit moving-root dependence. Artin continued-fraction digits eventually encode the cutoff and these rates excursion by excursion through individual digit degrees.
ER -