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Lemma 4.1 .
For T T large, over Q T ∖ P Q_{T}\setminus P , both ω ~ \tilde{\omega} and h h are positive. Moreover, h h has the following approximation formula
(4.16)
h \displaystyle h
= T 2 − 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 + 1 2 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 T T , and it has the singular behavior near P P given by Definition 3.3 .