AI 中文总结
该研究针对高阶单李群作用下的几乎平稳测度，发展量化理论并证明其二分法的有效形式，引入新工具并将其应用于离散子群内射半径的有效估计。
AI 中文摘要
设G是作用在空间X上的高阶单李群，Nevo和Zimmer的一个定理断言，X上的每个遍历平稳概率测度要么是G不变的，要么具有真抛物子群Q对应的射影因子G/Q。我们发展平稳测度的量化理论，证明该二分法的有效形式。引入ε-几乎平稳、δ-几乎不变和δ-几乎射影因子的概念，证明每个ε-几乎平稳测度要么是δ-几乎不变的，要么具有δ'-几乎射影因子，其中δ、δ'由ε显式给出且仅依赖于G。未施加遍历性、算术性或丢番图假设，且界对所有G空间一致。证明引入了若干工具：基于熵的鸽巢原理（entropigeonhole方法），它产生量化的Mautner现象；因子函数，即齐次因子空间上函数的量化类似物；以及群增长意义下的快速生成二分法。在配套论文中，这些工具被用于证明，无限协体积的离散子群在半径为r的球内某处的内射半径至少为c log^(4)r，这是Frączyk和Gelander定理的有效形式。
英文摘要
Let $G$ be a higher-rank simple Lie group acting on a space $X$. A theorem of Nevo and Zimmer asserts that every ergodic stationary probability measure on $X$ is either $G$-invariant or admits a projective factor $G/Q$ for a proper parabolic subgroup $Q$. We develop a quantitative theory of stationary measures and prove an effective form of this dichotomy. We introduce notions of $\eps$-almost stationarity, $δ$-almost invariance and $δ$-almost projective factor, and show that every $\eps$-almost stationary measure is either $δ$-almost invariant or carries a $δ'$-almost projective factor, with $δ,δ'$ explicit in $\eps$ and depending only on $G$. No ergodicity, arithmeticity or Diophantine hypothesis is imposed, and the bounds are uniform over all $G$-spaces. Additionally, we prove a decomposition version of this theorem where each measure is decomposed into an almost invariant part and an almost projective part. This comes at the cost of worse constants. The proof introduces several tools: the \emph{entropigeonhole method}, an entropy-based pigeonhole principle yielding a quantitative Mautner phenomenon; \emph{factor functions}, quantitative analogues of functions on homogeneous factor spaces; and a \emph{fast generation} dichotomy in the spirit of growth in groups. In a companion paper these are used to show, among other things, that a discrete subgroup of infinite covolume has injectivity radius at least $c\log^{(4)}r$ somewhere in the ball of radius $r$, which is an effective form of a theorem of Frączyk and Gelander. We also prove rates for Benjamini--Schramm convergence of quotients of higher-rank lattices.
