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Source. Theorem 1, p. 5, of Manjunath Krishnapur, Erik Lundberg and Koushik Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270v1 (24 March 2025), as identified on the source card.

Statement

Setting (p. 4). For compact $K\subseteq\mathbb C$, $\mathcal P_n(K)$ is the set of monic complex polynomials of degree $n$ with all zeros in $K$. For a polynomial $p$ and $t>0$, $\Lambda_p(t)={z\in\mathbb C:\lvert p(z)\rvert\le t}$ is its $t$-level lemniscate, and

$$\kappa_n(K,t)=\inf\{m(\Lambda_p(t)):p\in\mathcal P_n(K)\},$$

with $m$ the Lebesgue measure on the plane. $\mathbb T$ is the unit circle and $\overline{\mathbb D}$ the closed unit disc.

Theorem 1 (p. 5, quoted). "There exist $0<c<C<\infty$ such that for all large enough $n$,

$$\frac{c}{\log n}\leq\frac13\kappa_{n(\log n)^4}(\mathbb T,1)\leq\kappa_n(\overline{\mathbb D},1)\leq\kappa_n(\mathbb T,1)\leq\frac{C}{\log\log n}."$$

The middle comparison bounds the closed-disc problem below by the circle problem at the larger degree $n(\log n)^4$; the paper does not prove $\kappa_n(\overline{\mathbb D},1)=\kappa_n(\mathbb T,1)$ and calls that equality likely in Remark 4 (p. 5).

Proof pointer

The upper bound for $\kappa_n(\mathbb T,1)$ is proved in Section 4 (pp. 10--14) by a construction. The lower bound for zeros on the circle is proved in Section 6.1 (p. 19) from the theorem of Nazarov, Polterovich and Sodin on the area where a harmonic function is negative (the paper's Theorem 10, p. 7), applied to $\log\lvert p\rvert$. The comparison $\frac13\kappa_{n(\log n)^4}(\mathbb T,1)\le\kappa_n(\overline{\mathbb D},1)$ is the case $t=1$ of (31) (p. 25), proved as Corollary 25 in Section 7.3 (pp. 32--33) from the zero-pushing lemma (Lemma 21, Section 7.2) and the equidistribution of Theorem 7.

Read depth

Claims checked: the statement and the setting were read clause by clause on pp. 4--5 of the print; the proof was followed for its structure only.

Bears on

• Problem 116: the chain gives the paper's main Theorem (p. 2), whose lower bound $c/\log n$ is the problem's parenthetical $(\log n)^{-O(1)}$ form with exponent $1$, for all large $n$. The paper's footnote 1 (p. 2) says that since Theorem 1 compares the two constraints and the lower bound estimates the part of the lemniscate inside the unit disc, the results address both Erdős's 1940 version (zeros on the circle, area inside the disc) and the later closed-disc version.