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A solution to the Erd\H{o}s Problem #1040

Ioannis Tzachristas

arXiv:2609.06050v1math.CAcs.AI

TL;DR

The paper addresses whether compact sets of capacity one can have arbitrarily small unit-lemniscate area. It combines harmonic-polynomial positivity, controlled measure representation, and empirical approximation to prove that the area infimum vanishes without boundary regularity.

  • Problem

    The paper addresses whether the unit-lemniscate area infimum vanishes for compact sets of capacity one, the remaining compact case not covered by the earlier C2-boundary result.

    Source: §1 · pp. 1–2

  • Method

    The proof constructs a harmonic polynomial positive on most of the relevant set, represents it with a controlled signed measure, then perturbs and discretizes.

    Source: §1 · p. 2

  • Results

    For every a>0, a compact set K with cap(K)=1 admits a monic polynomial with zeros in K whose unit-lemniscate area is less than a.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The result proves the vanishing assertion for compact capacity-one sets without any regularity assumption.

    Source: §1 · p. 2

    The construction is existential, gives no degree bounds, explicit roots, or uniform decay rate, and recovers vanishing rather than the sharp exponential rate.

    Source: §7 · p. 7, §8 · p. 8

Abstract

from arXiv · show

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 Erd\H{o}s Problem 1040.

1 Introduction and statement of the result

For every compact set of logarithmic capacity one, the unit-lemniscate area infimum is zero. The proof combines harmonic-polynomial positivity, a controlled signed measure, and discretization to address the remaining capacity-one case without regularity assumptions.

1 Introduction and statement of the result

  • The paper addresses whether the infimum is determined by transfinite diameter and whether it vanishes for every closed infinite set with transfinite diameter at least one [1, Problem 4, p. 135].For compact sets, transfinite diameter equals logarithmic capacity; a counterexample to determination by capacity alone is reported by Feng et al. [2, Section 3.3].Source: §1 · p. 1
  • The threshold one cannot be lowered: below it, every admissible unit lemniscate contains a disk of fixed positive radius [1, Theorem 6, p. 135].Vanishing was previously proved when zeros lie on the unit circle [1, Theorem 4 and its corollary, p. 133].Source: §1 · p. 1
  • Earlier results covered capacity greater than one [3, Theorem 3.1] and capacity one for compact sets with C2 boundary [4, Theorem 6]; this paper proves the remaining compact case without regularity.The introduction of [3, version 3, 27 August 2026] left general compact sets of capacity one open.Source: §1 · p. 2
  • For compact K with cap(K)=1, every a>0 admits a monic polynomial with zeros in K whose unit-lemniscate area is less than a.Equivalently, ϑ(K)=0.Source: §1 · p. 2
  • The proof finds a centered harmonic function positive on most of the relevant set, represents it by a controlled signed measure, perturbs the equilibrium measure, and discretizes.The ingredients include a harmonic polynomial positive outside a thin corridor, a measure with bounded density relative to equilibrium measure, and L1 approximation of logarithmic potentials.Source: §1 · p. 2
  • The construction is existential, with no effective degree bound, explicit sequence of zeros, or uniform decay rate; the result extends to every closed infinite set of transfinite diameter at least one.The latter conclusion combines Theorem 1.1 with the capacity-greater-than-one and unbounded cases.Source: §1 · p. 2

2 Potential theory and harmonic-measure domination

This section sets up equilibrium measure and exterior harmonic measure, then establishes a domination bound that controls the measures used in the proof.

2 Potential theory and harmonic-measure domination

  • The logarithmic potentials used in the proof are locally integrable and belong to L1 on every bounded planar disk.The uniform integrability follows from integrability of the logarithmic singularity over a fixed disk.Source: §2 · p. 2
  • For compact nonpolar K, its polynomial hull H is compact and full, and the equilibrium probability measure μ is supported on ∂H⊂K.The equilibrium measures of H and K coincide, and cap(H)=cap(K).Source: §2 · pp. 2–3
  • For finite w outside H, ν_w is harmonic measure on ∂H, while harmonic measure at infinity is μ; exterior logarithmic balayage relates their potentials.The balayage formula requires no boundary smoothness and holds throughout H, including its interior.Source: §2 · p. 3
  • When H lies inside the disk of radius r and |w|=R>r, ν_w≤C_r,R μ; choosing R=2r gives ν_w≤3μ.The proof applies the disk Harnack inequality to the harmonic function w↦ν_w(A) for Borel A⊂∂H.Source: §2 · p. 3

3 Harmonic polynomials positive outside a thin corridor

Section 3 constructs a positive-area compact subset of the hull with arbitrarily small area loss and obtains a real harmonic polynomial associated with it. A Cauchy-transform extension contradiction establishes the polynomial-existence claim.

3 Harmonic polynomials positive outside a thin corridor

  • The construction yields a real harmonic polynomial h = Re(1 − P).The cited passage states this form for h; the surrounding bound is not included in the passage.Source: §3 · p. 3
  • The set E is formed by removing a thin neighborhood of a line segment, and it need not be full or polynomially convex.These distinctions avoid assumptions about continuation of Cµ through the original support.Source: §3 · p. 4
  • The compact set E can have positive area while m(H \ E) is less than both η and m(H)/2.The lemma applies to compact full H with positive area and any η > 0.Source: §3 · pp. 3–4
  • A Cauchy-transform extension contradiction completes the polynomial-existence claim.The argument uses the fact that a holomorphic extension across the selected boundary point would contradict its nonextension there.Source: §3 · p. 4

4 Realization by a controlled signed measure

Section 4 realizes a centered real harmonic polynomial as the logarithmic potential of a signed measure on K. Fourier averaging exterior harmonic measures provides the realization while controlling the measure relative to the equilibrium measure.

4 Realization by a controlled signed measure

  • A finite real signed measure σ on K has zero mass, bounded density relative to µ, and potential Uσ = h almost everywhere on H.The bound is |σ| ≤ Chµ, for a finite constant Ch.Source: §4 · p. 5
  • If h = 0, take σ = 0; otherwise choose R larger than the moduli of all points of H, and set wθ = Re^{iθ}; the logarithmic expansion converges uniformly for (z, θ) ∈ H × [0, 2π].Source: §4 · p. 5
  • The averaged measure is finite, has total mass zero, and is supported on ∂H ⊂ K.Its zero total mass follows from the zero mean of the averaging function b.Source: §4 · p. 5
  • The harmonic measures satisfy νwθ ≤ CRµ, which controls the total variation of the resulting signed measure.The constant CR is independent of θ.Source: §4 · p. 5

5 From probability measures to monic polynomials

Section 5 connects probability measures to monic polynomials through L1 convergence of logarithmic potentials and equally weighted empirical approximations. The area argument uses nonpositive sublevel sets and does not require control of the zero-level set's area.

5 From probability measures to monic polynomials

  • The planar-area approach applies to compact sets with irregular or positive-area boundaries.The passage contrasts this approach with relying on a measure on the boundary.Source: §5 · p. 6
  • Weak convergence of probability measures on compact K implies L1 convergence of their logarithmic potentials on every bounded disk.The proof truncates the logarithmic kernel, uses weak convergence for the truncated potentials, and then removes the truncation.Source: §5 · p. 6
  • Every probability measure on K can be approximated weakly by equally weighted empirical measures with denominators tending to infinity.The construction partitions K, approximates cell masses by rational numbers with a common denominator, and places the resulting multiplicities at points in the cells.Source: §5 · p. 6
  • For an empirical measure on roots zj,N, N^-1 log|pN| equals its logarithmic potential.The polynomial is the product of (z − zj,N), and the potential takes the value −∞ at its zeros.Source: §5 · p. 6
  • The area comparison bounds the portion of {UρN < 0} where Uρ > s by s^-1∥UρN − Uρ∥L1(B).The limiting set used is a nonpositive sublevel set; no area claim about the zero level set is required.Source: §5 · p. 6

6 Proof of the capacity-one theorem

The proof reduces the capacity-one case to making the potential’s nonpositive set arbitrarily small in planar area. A positive perturbation of the equilibrium measure and empirical approximation then give the vanishing result.

6 Proof of the capacity-one theorem

  • When the polynomial hull H has zero area, the equilibrium potential is positive outside H, so Proposition 5.2 gives ϑ(K) = 0 immediately.This handles the zero-area hull case directly.Source: §6 · p. 7
  • For m(H) > 0, a real harmonic polynomial and a signed measure σ with bounded density relative to µ provide the perturbation used in the proof.The construction chooses σ with σ(K) = 0 and Uσ = h almost everywhere on H.Source: §6 · p. 7
  • The perturbed measure ρt remains a probability supported on K, with density relative to µ greater than 1/2.Its potential outside the hull is controlled using Uµ(z) = gΩ(z, ∞) > 0.Source: §6 · p. 7
  • Choosing t sufficiently small makes m{Uρt ≤ 0} < 2η for arbitrary η > 0.The proof controls the exterior contribution using boundedness of the nonpositive sublevel sets and dominated convergence.Source: §6 · p. 7
  • Since Proposition 5.2 gives ϑ(K) < 2η and η is arbitrary, ϑ(K) = 0, with empirical approximation yielding a polynomial of area less than any prescribed a > 0.The approximation is taken far enough along the empirical sequence after choosing 2η < a.Source: §6 · p. 7

7 Completion of the vanishing question

The paper completes the vanishing question by combining the compact capacity-one result with the capacity-greater-than-one case and an argument for unbounded sets. Thus every closed infinite set of transfinite diameter at least one has vanishing infimum.

7 Completion of the vanishing question

  • For compact K with cap(K) > 1, ϑ(K) = 0 by the positive equilibrium potential and Proposition 5.2.The argument here recovers vanishing, while the sharp exponential decay rate in [3, Theorem 3.3] is stronger.Source: §7 · p. 7
  • For unbounded F, choosing two points at distance D > 2 gives a degree-two polynomial whose unit-lemniscate area is at most 2π(D − 1)^−2.Letting D tend to infinity proves ϑ(F) = 0.Source: §7 · pp. 7–8
  • Every closed infinite F with transfinite diameter at least one satisfies ϑ(F) = 0, by applying the compact results when bounded and the unbounded argument otherwise.Source: §7 · p. 8
  • The unbounded argument does not require a compact subset of capacity greater than one, an assumption that can fail for unbounded closed discrete sets.Source: §7 · p. 8

8 Further remarks

The capacity-one theorem removes the boundary regularity assumption from the earlier result of Krishnapur, Lundberg, and Ramachandran [4, Theorem 6]. The paper also distinguishes the vanishing question from whether the infimum is determined by transfinite diameter alone.

8 Further remarks

  • Theorem 1.1 proves the capacity-one result without boundary regularity, using L1 convergence of potentials rather than uniform convergence at logarithmic singularities.The broad perturbation strategy comes from [4], while Sections 3–5 supply arguments for arbitrary compact sets.Source: §8 · p. 8
  • The construction is existential: it provides no effective degree bound, explicit sequence of zeros, or uniform decay rate in the degree.The order of choices is essential, and intermediate constants are not asserted to be uniform as the corridor narrows.Source: §8 · p. 8
  • The vanishing question has an affirmative answer [1, Problem 4], but determination by transfinite diameter alone has a negative answer [2, Section 3.3] and [3, Example 2.1].Source: §8 · p. 8

A A normalization check on the unit circle

The unit-circle example checks the sign and factor in the representation lemma for a centered harmonic polynomial. It does not itself yield arbitrarily small area.

A A normalization check on the unit circle

AI assistance and review status

The proposed result was developed with GPT-6-Astra through Codex, with AI assistance spanning proof work and manuscript review. Internal agent checks were not independent human verification, and the draft asserts neither formalization nor external mathematical endorsement.

AI assistance and review status

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