There are some uniform constants $\underline{v}_{0}>0$ and $\overline{v}_{0}>0$ independent of $T\gg 1$ such that for each $\bm{x}\in\mathcal{M}_{T}$, we have

(4.270)

$$\underline{v}_{0}\leq\frac{\mathfrak{s}(\bm{y}_{1})}{\mathfrak{s}(\bm{y}_{2})}\leq\overline{v}_{0}$$

for all $\bm{y}_{1},\bm{y}_{2}\in B_{\frac{\mathfrak{s}(\bm{x})}{4}}(\bm{x})$.