A solution to the Erdős Problem #1040

• URL: http://arxiv.org/abs/2609.06050v1
• Date: Sat, 05 Sep 2026 12:11:52 GMT
• Title: A solution to the Erdős Problem #1040
• Abstract summary: We prove that $vartheta(K)$ for every closed infinite set $FsubsetmathbbC$ transfinite diameter at least one, answering the vanishing question in Erds Problem 1040.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$. A Fourier average of exterior harmonic measures realizes this polynomial as the logarithmic potential of a signed measure having bounded density with respect to the equilibrium measure. A positive perturbation and an $L^1$ approximation by empirical measures then produce the required polynomials. This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran to arbitrary compact sets of capacity one. Together with the capacity-greater-than-one theorem of Ghosh and Ramachandran and an elementary argument for unbounded sets, it follows that $\vartheta(F)=0$ for every closed infinite set $F\subset\mathbb{C}$ of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040.

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