distance $\sigma_{n_0}$ or more away. Inductively one can show that there are at most $4^{m-n_0+k}$ terms having singularities a distance $\sigma_{n_0-k}$ away from $z$, and so Schwarz lemma bounds the sum of these terms by a constant multiple of $$\begin{aligned}
    \frac1{m^3}\sum_{k=0}^{n_0}\frac{\sigma_m^2\delta_m^2 4^{m-n_0+k}}{\sigma_{n_0-k}^2}
        &\le& \frac1{m^3}\sum_{k=0}^{n_0}\frac{4^m\sigma_m^2}{4^{n_0-k}\sigma_{n_0-k}^2}\le\frac1{m^3}\sum^{n_0}_{k=0}1  \\
        &\le& \frac{n_0}{m^3}\le\frac1{m^2}
    \end{aligned}$$ Similarly, if $m<n_0$, there are $4^{k-m}$ terms at roughly distance $\sigma_{m-k}$, and so for these terms the sum is controlled by $m^{-3}\sum_0^m\frac{\sigma_m^24^{k-m}}{\sigma_{m-k}^2}\lesssim m^{-2}$. Summing these bounds over $m$, we obtain $\Vert\sum_{n,J,j}h^n_{J,j}\Vert_{L^\infty(\Omega)}\lesssim1$ as desired. This proves the theorem.\

Proof of Proposition \[basicconstruction\]
==========================================

Let us now turn to the proof of Proposition \[basicconstruction\]. By normal families, it suffices to perform the construction in the domains $\{\hat T^n_J:|J|=n,n\le N\}$, provided we obtain constants independent of $N$.

Rather than working directly with the sets $\hat T^n_J$, we will consider slightly enlarged domains $\tilde T^n_J$, defined as follows. We first define $\tilde T^N_J=\hat T^N_J$. Now let $\phi^N_J$ be a conformal mapping from $\tilde T^N_J$ to $\mathbb D$ preserving the symmetries of the domain $\tilde T^N_J$. Let $E_j=E_j(N,J)$ be the arc of $\partial\mathbb D$ defined by $E_j:=\phi^N_J(\partial L^N_{J,j}\cap\partial\hat T^N_J)$, and let $E_j^\ast=3E_j$. Defining $\theta_N=\pi(1-\frac{c_1}{\log N})$, the constant $c_1$ to be determined later, let $\gamma_j^\ast$ be the circular 