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Claim. There are absolute constants $c,C>0$ such that for every $n\ge3$,

$$\frac{c}{\log n}\le\inf_p\,\lvert\{z:\lvert p(z)\rvert<1\}\rvert\le\frac{C}{\log\log n},$$

the infimum over monic $p$ of degree $n$ with all zeros in the closed unit disk. This is the main theorem of Krishnapur, Lundberg and Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270 (2025-03-24, 44 pages), digested on the card [[../library/polynomials/krishnapur_2025_area_polynomial_lemniscates/_index|Krishnapur, Lundberg and Ramachandran 2025]]. The lower bound answers the stronger, parenthetical form of Problem 116, an area of at least $(\log n)^{-O(1)}$, with exponent $1$, and so also the $n^{-O(1)}$ form that Pommerenke 1961 settled with exponent $4$; the upper bound sharpens Wagner's construction [Wa88], whose area was $\ll_\varepsilon(\log\log n)^{-1/2+\varepsilon}$, and leaves a gap between $1/\log n$ and $1/\log\log n$, which Pendyala's 2026 preprint (arXiv:2606.17097) claims to close at order $1/\log n$. The paper derives the bounds from a finer theorem relating the closed-disk constraint to zeros on the unit circle, with potential theory, equilibrium measures and a probabilistic construction; it also gives the sharp order $(\log\log n)^{-1}$ for the sublevel sets at every level above $1$ and an inradius bound of order $1/(n\sqrt{\log n})$, which bears on Problem 1039.

Depends on. No page of this wiki.

Acceptance. Reviewed: erdosproblems.com labels the problem proved and states the $(\log n)^{-1}$ lower bound and the $(\log\log n)^{-1}$ upper bound as proved by this paper, cited as [KLR25] (page last edited 2025-10-24), which the corpus counts as documented independent acceptance by the site's curator, T. F. Bloom (erdosproblems.com); the formal-conjectures file lists the logarithmic form as solved without a formal proof, so the evidence lists no formalized kind. Not refereed: the preprint is at its first arXiv version and no journal version is recorded. Proof coverage: the card's digest only; the proof is unverified by this corpus, and the standing rests on the site's acceptance.