ScalingStacks

Lemma 4.1 . [051G]

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Lemma 4.1.

For TT large, over QT∖PQ_{T}\setminus P, both ω~\tilde{\omega} and hh are positive. Moreover, hh has the following approximation formula

(4.16) h\displaystyle h =T2−n​(T+k±​z)n−1+ϵ⁡(z),|z|≥1,\displaystyle=T^{2-n}(T+k_{\pm}z)^{n-1}+\epsilon(z),\quad|z|\geq 1,
(4.17) h\displaystyle h =T+12​r+O′​(r)+T−1​B​(z),|z|≤1,\displaystyle=T+\frac{1}{2r}+O^{\prime}(r)+T^{-1}B(z),\quad|z|\leq 1,

where O′​(r)O^{\prime}(r) is a fixed function independent of TT, and it has the singular behavior near PP given by Definition 3.3.

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