ScalingStacks

Proof of Theorem 2.3. We will first show (2.10), which is an easy consequence of (2.9). The left-hand side follows immediately from (2.9), which implies

(3.27)

$$\textrm{tr}_{\tilde{\omega}_{y}}\omega_{y}\leq\frac{C}{t}e^{Ce^{B\sigma^{-\lambda}}}.$$

Then (3.21) and (3.7) give

(3.28)

$$\textrm{tr}_{\omega_{y}}\tilde{\omega}_{y}\leq(\textrm{tr}_{\tilde{\omega}_{y}}\omega_{y})^{n-m-1}\frac{\tilde{\omega}_{y}^{n-m}}{\omega_{y}^{n-m}}\leq t\frac{Ce^{Ce^{B\sigma^{-\lambda}}}}{\sigma^{\lambda}}\leq tCe^{Ce^{B\sigma^{-\lambda}}},$$

which proves (2.10).