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Source. Theorem 1.1, p. 1, proof pp. 2--4 (Section 2), of Linhang Huang, Many lemniscates with large diameter, arXiv:2509.11597 (2025), version 2, the edition named on the source card.

Statement

Theorem 1.1 (p. 1, quoted). "For each $c \in (0, 4)$ and $N \in \mathbb{N}$, there exists a monic polynomial $p(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0$ such that $\{z \in \mathbb{C} : |p(z)| \le 1\}$ has at least $N$ connected components with diameter at least $c$."

The theorem places no bound on the degree $n$. The set is the closed sublevel set $|p|\le1$.

Sharpness, as the paper reports it (p. 1). The paper calls the restriction $c<4$ best possible, citing Pólya (1928): for a monic polynomial $p$, the orthogonal projection of $\{|p|\le1\}$ onto any line can be covered by intervals of total length at most $4$. This is Pólya's theorem, cited and not proved in the paper. The paper adds (p. 2) that a segment of length $\ell$ has logarithmic capacity $\ell/4$.

Read depth. Claims checked: the statement was read clause by clause on p. 1 of the version 2 PDF, and the proof of Section 2 (pp. 2--4) was read through once. Nothing here is independently reviewed.

Proof pointer

Section 2, pp. 2--4, outlined here.

1. For $0<c<4$, the shifted Joukowski map $\varphi(z)=\tfrac c4\bigl(z+\tfrac1z+2\bigr)$ maps the exterior of the closed unit disc onto the complement of $[0,c]$. The domain $\Omega$ bounded by the image of the circle $|z|=4/c$ contains $[0,c]$, and the rescaled map $\varphi(4z/c)=z+O(1)$ shows that $\Omega$ has logarithmic capacity $1$ (Section 2.1, pp. 2--3).
2. Inside $\Omega$ take $N$ pairwise disjoint Jordan curves, any two separated by a positive distance and none touching $\partial\Omega$, each bounding a domain of diameter greater than $c$; the union of these domains is $\Omega_N$ (Section 2.2, p. 3).
3. The Hilbert Lemniscate Theorem, in the form of Bloom, Levenberg and Lyubarskii, gives a polynomial $q$ whose sublevel set at level $\sup_{\Omega_N}|q|$ contains $\Omega_N$ and lies in a small neighbourhood of $\Omega_N$ inside $\Omega$, so that set has at least $N$ components of diameter at least $c$. After normalising so that this level is $1$, the identity $\mathrm{Cap}(r^{-1}(\overline{\mathbb D}))=|a_d|^{-1/d}$ for a polynomial $r$ of degree $d$ with leading coefficient $a_d$ (Ransford, Theorem 5.2.5) and monotonicity of capacity give $|a_d|\ge1$ for $q$. Setting $p(z)=q(z/w)$ with $w^d=a_d$ makes $p$ monic, and since $|w|\ge1$ the components are only enlarged (Section 2.3, p. 4).

Dependencies

The Hilbert Lemniscate Theorem (Hilbert; the formulation of Bloom, Levenberg and Lyubarskii, Ann. Inst. Fourier 58 (2008)), the description of logarithmic capacity of a continuum through the exterior Riemann map, and Ransford, Potential theory in the complex plane, Theorem 5.2.5. All are cited, not proved, in the paper.

Bears on

• Problem 511: the paper states (p. 1) that the theorem answers the question of Erdős, which it identifies as problem #511, whether the number of connected components of $\{|p|\le1\}$ with diameter greater than $1+c$ is bounded by a constant $A(c)$ independent of the degree. The theorem is stated for the closed set $\{|p|\le1\}$ and diameters at least $c$; the problem's statement uses the strict inequality $|f|<1$ and diameters greater than $c$, for $c>1$. The paper's note added (p. 2) says the problem had already been solved by Pommerenke (Michigan Math. J. 8 (1961)), and calls its own proof an independent rediscovery.