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Manjunath Krishnapur, Erik Lundberg and Koushik Ramachandran, Inradius of random lemniscates, J. Approx. Theory 299 (2024), Paper No. 106018 (DOI registered 3 February 2024, issue of May 2024), prove the second question for regular compact sets of capacity above $1$. Their Corollary 1.6, as Ghosh and Ramachandran restate it (arXiv:2604.03036v3, p. 2), states that a compact $K$ with $\operatorname{cap}(K)>1$ has $\mu(K)=0$ provided there are constants $C,\eta>0$ such that the equilibrium measure $\nu$ of $K$ satisfies $\nu(B(z,r))\le Cr^\eta$ for all discs $B(z,r)$. The authors' later paper records the stronger form $\kappa_n(K,1)\le e^{-cn}$ for some $c>0$ depending on $K$, where $\kappa_n(K,1)$ is the minimal area of $\{z:|p(z)|<1\}$ over monic $p$ of degree $n$ with all zeros in $K$, and calls the result an affirmative answer when the capacity exceeds one. The statement here is given as those two papers restate it. The arXiv v1 of 31 January 2023 has no Corollary 1.6: its Corollary 1.4, under its assumption (D) and a positive lower bound on the potential, is the underlying result on random zeros, and the label belongs to the journal version only. The proof is not checked here.

Covers. The second question for compact $K$ of capacity above $1$ whose equilibrium measure satisfies such a bound; nothing about capacity exactly $1$, unbounded sets or the first question. The case is wholly contained in the theorem on [[problems/analysis/E1040/claims/2026_04_03_ghosh_ramachandran|the page of Ghosh and Ramachandran]], which needs no regularity assumption.

Refereed. J. Approx. Theory 299 (2024), Paper No. 106018. The site does not name the paper.

Depends on. Nothing in this wiki.