A solution to the Erdős Problem #1040

Abstract

For a compact set K⊂ℂ, let ϑ(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 ϑ(K)=0 whenever 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 L¹ 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 ϑ(F)=0 for every closed infinite set F⊂ℂ of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040.

On the solution to the Erdős-Hajnal problem on high-girth high-chromatic subgraphs
Tung Nguyen, Bartosz Walczak
arXiv1 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
A solution to the Erdős-Sárközy product-density problem
John M. Campbell
arXiv1 Nov 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Linear dependencies, polynomial factors in the Duke–Erdős forbidden sunflower problem – CORRIGENDUM
Andrey Kupavskii, Fedor Noskov
Forum of Mathematics, Sigma29 Oct 2026Peer-reviewedCC BY 4.0General mathematicsCrossref
On the Erdős--Ginzburg--Ziv Problem for ℤ_n^r
Dongchun Han, Hanbin Zhang
arXiv29 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
On a problem of Erdős and Nathanson related to minimal asymptotic bases of order h
Shi-Qiang Chen, Quan-Hui Yang
arXiv8 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{tzachristas2026solution,
  title = {{A solution to the Erdős Problem #1040}},
  author = {Ioannis Tzachristas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06050},
  eprint = {2609.06050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06050},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A solution to the Erdős Problem #1040
AU  - Ioannis Tzachristas
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06050
UR  - https://arxiv.org/abs/2609.06050
AB  - For a compact set K⊂ℂ, let ϑ(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 ϑ(K)=0 whenever 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 ϑ(F)=0 for every closed infinite set F⊂ℂ of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040.
ER  -

Abstract

Related

Cite