Frączyk-Gelander proved in 2024 that arbitrarily large balls can be embedded into any higher rank infinite volume symmetric space. In a recent work with Ilya Gektman, Simon Machado and Omri Solan we quantify their result, giving an explicit growth rate for the maximal injectivity radius. More precisely, let G be a higher rank simple Lie group and let \Gamma<G be a discrete non-lattice subgroup. Then for some c=c(Gamma) and for every R>0, there is some g\in G_{clog log log(R)} (the ball of radius log log log R in the Riemannian metric) such that a ball of radius R can be injected into G/\Gamma centered at g\Gamma. Along the way we give a new simpler proof of the Stuck-Zimmer theorem which doesn’t use the intermediate factor theorem and in addition yields a quantitative version of the Stuck-Zimmer theorem. Our methods invoke the novel concepts of almost stationary and almost invariant measures and a corresponding new quantitative version of the celebrated Nevo-Zimmer Structure Theorem for higher rank stationary actions. In addition, we analyze almost invariant measures on G/\Gamma using a new characterization of lattices which says, roughly, that if G/\Gamma supports an almost invariant measure then \Gamma is a lattice. This analysis uses the work of Bader, Furman, Gelander and Monod about property (T) for actions on Banach spaces.