First of all note that since $X(U_{v}^{\beta})\cap Z_{a}\subset(\mathbb{C}\backslash\{0\})^{d}$, both $\log|z|$ and $\operatorname{Arg}(z)$ are well defined. Also by (2) of Lemma 3.2 the values of the vector field $\mathfrak{X}$ are in $(\operatorname{carrier}v)^{\vee}$. Hence, for any ${q\in U_{v}^{\beta}}$ the ray $\mathcal{F}_{q}$ is in $Q^{\lambda}_{(v|\{0\})}(\beta)$ and the standard estimates on values of the monomials at $z\in\log^{-1}(Q^{\lambda}_{(v|\{0\})}(\beta))$ apply:

$$|a_{m}z^{m}|\leq e^{-\beta}|a_{v}z^{v}|,\text{ all }m\neq v,\{0\}.$$

Or putting them all together we have

$$\displaystyle\left|\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right|\leq|\Delta_{\mathbb{Z}}|e^{-\beta}.$$

Hence,

$$\displaystyle\left|\log\left(1+\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right)\right|\leq C^{\prime}(\beta)\cdot|\Delta_{\mathbb{Z}}|e^{-\beta},$$

where $C^{\prime}(\beta)\to 1$ as $\beta\to\infty$. Writing the equation of $Z_{a}$ in $X(U_{v}^{\beta})$ as

$$a_{v}z^{v}\left(1+\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right)=1$$

or, equivalently,

$$\log(a_{v}z^{v})=-\log\left(1+\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right)$$

will give the claimed estimates. ∎