ScalingStacks

Remark 4.5. We discuss some intuition about log scales. Let $P\in X_{s}$ lie in $U_{w,\delta}^{s}$, then a log scale $|z^{m_{i}}|\sim|z^{m_{i}}(P)|$ around $P$ refers to the subregion

$$\{\frac{1}{2}|z^{m_{i}}(P)|\lesssim|z^{m_{i}}|\lesssim 2|z^{m_{i}}(P)|,\quad 1\leq i\leq n\}.$$

Now $\log|z^{m_{i}}|$ vary by order $O(s)$ within $U_{w,\delta}^{s}$, so there are an enormous number of log scales. The long range behaviour of $X_{s}$ is similar to $(\mathbb{C}^{*})^{n}$, with half of the dimensions compactified into $T^{n}$. On the other hand, over one log scale $X_{s}$ behaves qualitatively like the unit disc in $\mathbb{C}^{n}$. The concept of local oscillation of a function refers to the oscillation within one log scale. In particular the Lipschitz bound (23) implies a local oscillation bound

$$\text{osc}_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}\bar{\phi}\leq Cs^{-1}.$$