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A solution to the Erdos Problem #1040
arXiv:2609.06050v1 Announce Type: cross Abstract: For a compact set KsubsetC, let artheta(K) be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in K, allowi
arXiv:2609.06050v1 Announce Type: cross Abstract: For a compact set KsubsetC, let artheta(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 artheta(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 artheta(F)=0 for every closed infinite set FsubsetC of transfinite diameter at least one, answering the vanishing question in Erdos Problem 1040.
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Source: arXiv cs.AI | 2026-09-10