AI 中文总结
本文证明实半单群中弱内射半径性质等价于球面密度假设，通过谱壁均匀估计和正预迹核实现，无需温谱或谱隙条件。
AI 中文摘要
设 $G$ 为具有有限中心的连通半单实李群，且 $(\Gamma_N)$ 为余紧格子的塔。我们证明弱内射半径性质蕴含具有相同参数的球面密度假设。结合 Golubev 和 Kamber 的逆定理，这给出了余紧阿基米德情形下的等价性。证明使用了谱壁附近的均匀球面质量估计和正预迹核，但不需要温谱估计或先验谱隙。
英文摘要
Let $G$ be a connected semisimple real Lie group with finite center and let $(Γ_N)$ be a tower of cocompact lattices. We prove that the weak injective radius property implies the spherical density hypothesis with the same parameter. Together with the converse theorem of Golubev and Kamber, this gives an equivalence in the cocompact Archimedean setting. The proof uses a uniform spherical-mass estimate near spectral walls and a positive pre-trace kernel, but no tempered-spectrum estimate or a priori spectral gap.
Comments18 pages
