发表机构
Catholic Kwandong University（嘉陵江天主教韩国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究长商格上沿特定子群的右\(K -\)球面平均，通过\(K -\)球面投影转化动力族为树问题，证明精确偏差公式，给出紧致与稠密轨道的\(K -\)球面等分布情况，紧致轨道偏差为零，稠密情况速率受边界点连分数展开控制。
AI 中文摘要
设\(F = \mathbb{F}_q(\!(t^{-1})\!)\)，\(G = \mathrm{SL}_2(F)\)，\(\Gamma = \mathrm{SL}_2(\mathbb{F}_q[t])\)，\(X = \Gamma\backslash G\)，且\(K = \mathrm{SL}_2(\mathcal{O})\)，其中\(\mathcal{O} = \mathbb{F}_q[\![t^{-1}]\!]\)。研究了长商格上沿上幂单子群（与标准尖点相关的水平球子群）的右\(K -\)球面平均。基本观察是\(K -\)球面投影将两个自然动力族转化为布鲁哈特 - 蒂茨树上相同的有根后代问题。在偶二分扇区，极限高度律是明确的概率测度。证明了精确偏差公式，给出了紧致\(U -\)轨道扩张平移和稠密轨道截断的定量\(K -\)球面等分布情况。对于紧致轨道扩张平移中紧致支撑的\(K -\)球面可观测量，偏差最终恰好为零。在稠密情况下，速率由轨道附着的边界点的连分数展开控制。
英文摘要
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.
Comments31 pages, 6 figures
