Lemma 16.28.50.C78FAC45EABB The set $E$ is compact, and every neighborhood of each of its points contains an open disk contained in $E$. The series
\[ h(\zeta )=\sum _{j\geq 1}2^{-j}\log |\zeta -p_ j|+2\log 2 \]
defines a subharmonic function on $\mathbb C$ of logarithmic growth at most one; it satisfies $h(2)=0$ and is at most $-16+2\log 2$ on every small disk $\overline D(p_ j,r_ j)$. The function
\[ v(\zeta )=\frac{8\log ^+|\zeta |+h(\zeta )-4\log 2}{9} \]
is subharmonic of logarithmic growth at most one, is nonpositive on $E\setminus \{ 2\} $, and satisfies $v(2)=4\log 2/9$.
Proof.
Since $r_ j<2^{-j-3}$, the small disks are pairwise disjoint and disjoint from $\overline D$, and they accumulate only at $2$. This proves compactness. Every neighborhood of $2$ contains one of the small disks, and every other point of $E$ lies in the closure of the interior of one of the disks.
To show that the series defines a subharmonic function, write $\chi (\zeta )=\log (1+|\zeta |^2)$ on $\mathbb C$ and $\theta =\omega _{\mathrm{FS}}/2$ on $\mathbb P^1$. By lemma 16.28.9, each
\[ q_ j=\log |\zeta -p_ j|-\chi /2 \]
extends to a $\theta $-plurisubharmonic function on $\mathbb P^1$. The Cauchy–Schwarz inequality gives $q_ j\leq \log 4$, while the value of $q_ j$ at infinity is zero. Thus $0\leq \sup q_ j\leq \log 4$, and theorem 5.25.2 bounds $\| q_ j-\log 4\| _{L^1}$ uniformly. The partial sums
\[ w_ N=\sum _{j=1}^ N2^{-j}(q_ j-\log 4) \]
decrease, remain $\theta $-plurisubharmonic and have uniformly bounded $L^1$ norms. By corollary 5.25.3, their limit $w$ belongs to $\mathrm{PSH}(\mathbb P^1,\theta )$. Since $\sum _ j2^{-j}=1$,
\[ h=w+\chi /2+\log 4+2\log 2 \]
on the affine chart. This proves subharmonicity and the growth bound, and $\sum _ j j2^{-j}=2$ gives $h(2)=0$.
On the $j$th small disk, all distances to the other centers are less than one, so the corresponding terms are negative, and
\[ h\leq 2^{-j}\log r_ j+2\log 2=-16+2\log 2. \]
On $\overline D$ all distances to the centers are less than four, hence $h\leq 4\log 2$ and $v\leq 0$. On each small disk $|\zeta |<4$, so $9v\leq 14\log 2-16<0$. The growth bound for $v$ and its value at $2$ follow directly from the formula.
$\square$
Pome
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