The improved Skoda estimate is one of the main discoveries in [52]. Intuitively, this means $\phi-\bar{\phi}$ can only be significantly below $-\frac{Const}{|\log|t||^{1/2}}$ on sets with exponentially small measure. Compounded with the upper bound $\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$, this means for sufficiently small $t$, an arbitrary bounded psh function $\phi$ is very close to the convex function $\bar{\phi}$ except on exponentially small measure.