$K$-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy

Abstract

Let $q$ be a prime power, $F=\mathbb F_q(\!(t^{-1})\!)$, $G=\mathrm{SL}_2(F)$, $Γ=\mathrm{SL}_2(\mathbb F_q[t])$, and $K=\mathrm{SL}_2(\mathbb F_q[\![t^{-1}]\!])$, and let $U<G$ be the upper unipotent subgroup. We study right $K$-spherical averages along $U$ on $X=Γ\backslash G$. Expanding translates of compact $U$-orbits and compact-open F$\unicode{x00F8}$lner-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 F$\unicode{x00F8}$lner-ball averages. For a depth-$N$ shadow rooted at height $k$, with cutoff $M=N-k\ge0$, 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 $|Φ(2m)|\le Cq^{αm}$, $α<2$, 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.