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

Abstract

Let $q$ be a prime power, $F={\mathbb{F}}_q((t^{-1}))$, $G={\mathrm{SL}}_2(F)$, $\Gamma ={\mathrm{SL}}_2({\mathbb{F}}_q[t])$, 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 $X=\Gamma \backslash G$. 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 $M=N-k\ge 0$, 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 $|\Phi (2m)|\le C{q}^{\alpha m}$, $\alpha <2$, three rate regimes arise, with a linear-in-scale factor at $\alpha =1$ and explicit moving-root dependence. Artin continued-fraction digits eventually encode the cutoff and these rates excursion by excursion through individual digit degrees.

31 pages, 6 figures