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A solution to the Erdős Problem #1040
Ioannis Tzachristas
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 9/9/2026, 5:33:27 AM
Summary
This paper solves Erdős Problem 1040 by proving that the infimum of the planar areas of unit lemniscates, denoted ϑ(K), is zero for any compact set K in the complex plane with logarithmic capacity equal to one. The proof utilizes harmonic polynomials, exterior harmonic measures, and L1 approximation by empirical measures, extending previous results that required smooth boundary regularity. Combined with existing theorems for capacity greater than one and unbounded sets, the paper establishes that ϑ(F)=0 for every closed infinite set F of transfinite diameter at least one.
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Ioannis Tzachristas → authored → A solution to the Erdős Problem #1040
confidence 100% · A solution to the Erdős Problem #1040 Ioannis Tzachristas
A solution to the Erdős Problem #1040 → solves → Erdős Problem 1040
confidence 100% · answering the vanishing question in Erdős Problem 1040.
Krishnapur → collaboratedwith → Lundberg
confidence 95% · This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran
Krishnapur → collaboratedwith → Ramachandran
confidence 95% · This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran
Lundberg → collaboratedwith → Ramachandran
confidence 95% · This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran
Ghosh → collaboratedwith → Ramachandran
confidence 95% · Together with the capacity-greater-than-one theorem of Ghosh and Ramachandran
Erdős → proposed → Erdős Problem 1040
confidence 95% · Erdős, Herzog, and Piranian asked whether this infimum is determined by transfinite diameter... These questions are recorded as Erdős problem 1040.
Herzog → proposed → Erdős Problem 1040
confidence 90% · Erdős, Herzog, and Piranian asked whether this infimum is determined by transfinite diameter... These questions are recorded as Erdős problem 1040.
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Abstract
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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- Source: https://arxiv.org/abs/2609.06050v1
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A solution to the Erdős Problem #1040 Ioannis Tzachristas 5 September 2026 Abstract For a compact set K⊂ℂK , let ϑ(K) (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 (K)=0 whenever cap(K)=1 (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 L1L^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 (F)=0 for every closed infinite set F⊂ℂF of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040. Keywords. Polynomial lemniscates; logarithmic capacity; equilibrium measure; balayage; harmonic measure. 2020 Mathematics Subject Classification. 30C10, 31A15. 1 Introduction and statement of the result Write m for planar Lebesgue measure. For a nonempty set F⊂ℂF , define ϑ(F)=infn≥1infz1,…,zn∈Fm(z∈ℂ:|∏j=1n(z−zj)|<1). (F)= _n≥ 1\ _z_1,…,z_n∈ Fm ( \z : | _j=1^n(z-z_j) |<1 \ ). (1.1) The points zjz_j need not be distinct. In particular, every polynomial in this definition is monic and has all its zeros in the prescribed set. Erdős, Herzog, and Piranian asked whether this infimum is determined by transfinite diameter, and in particular whether it vanishes for every closed infinite set of transfinite diameter at least one [1, Problem 4, p. 135]. For compact sets, transfinite diameter equals logarithmic capacity; we use the notation cap for either quantity. These questions are recorded as Erdős problem 1040. They have distinct answers. A counterexample to determination by capacity alone is reported by Feng et al. [2, Section 3.3]. Ghosh and Ramachandran [3, Example 2.1] give compact sets with different values of ϑ at every prescribed capacity in (0,1)(0,1). The present paper addresses the vanishing question at and above capacity one. Erdős, Herzog, and Piranian already proved vanishing when the allowed zeros lie on the unit circle [1, Theorem 4 and its corollary, p. 133]. They also proved that for a closed set of transfinite diameter less than one every admissible unit lemniscate contains a disk of a fixed positive radius [1, Theorem 6, p. 135]. In particular, the threshold one in the vanishing question cannot be lowered. The capacity-greater-than-one case was proved without regularity assumptions by Ghosh and Ramachandran [3, Theorem 3.1]. They also establish the sharp exponential decay rate [3, Theorem 3.3]. At capacity one, Krishnapur, Lundberg, and Ramachandran prove the vanishing assertion when K is the closure of a bounded open set with C2C^2 boundary [4, Theorem 6]. The introduction of [3, version 3, 27 August 2026] explicitly leaves general compact sets of capacity one open. We prove the remaining compact case without any regularity assumption. Theorem 1.1. Let K⊂ℂK be compact with cap(K)=1 (K)=1. For every a>0a>0 there are an integer n≥1n≥ 1 and points z1,…,zn∈Kz_1,…,z_n∈ K such that the monic polynomial p(z)=∏j=1n(z−zj)p(z)= _j=1^n(z-z_j) satisfies mz∈ℂ:|p(z)|<1<a.m\z :|p(z)|<1\<a. Equivalently, ϑ(K)=0 (K)=0. The proof follows the broad strategy of [4, Section 5]: find a centered harmonic function positive on most of the relevant set, represent it by a controlled signed measure, perturb the equilibrium measure, and discretize. To handle arbitrary compact sets, we use three ingredients. First, a Cauchy-transform singularity and Hahn–Banach argument give a harmonic polynomial with the required positivity outside a thin corridor. Second, averaging exterior harmonic measures realizes this polynomial by a measure whose density is bounded relative to the equilibrium measure, without assumptions on the boundary. Third, convergence of logarithmic potentials in planar L1L^1 replaces geometric discretization of a smooth boundary. All three ingredients are proved below. The construction is existential. It gives no effective degree bound, explicit sequence of zeros, or uniform rate of decay in the degree. Section 7 combines Theorem 1.1 with the capacity-greater-than-one case and the unbounded case to prove Corollary 7.3: every closed infinite set of transfinite diameter at least one has ϑ(F)=0 (F)=0. 2 Potential theory and harmonic-measure domination Write (K)P(K) for the Borel probability measures supported on K. For a compactly supported finite positive measure ρ, use the positive-log convention Uρ(z)=∫log|z−t|ρ(t).U_ρ(z)= |z-t|\,dρ(t). (2.1) The same notation is used for a finite signed measure wherever its positive and negative parts have finite potentials, and elsewhere as an almost-everywhere defined locally integrable function. On any bounded planar disk B, for t in a fixed compact set K, supt∈K∫B|log|z−t||m(z)<∞. _t∈ K _B| |z-t||\,dm(z)<∞. (2.2) Indeed, after translating by t, the integration region lies in one fixed disk, and the logarithmic singularity is integrable there. Consequently all the potentials used below belong to L1(B)L^1(B). Let K be compact and nonpolar, let H=K^H= K be its polynomial hull, and put Ω=ℂ∖H =C H. Thus H is K together with the bounded components of ℂ∖KC K; it is compact and full, meaning that its complement is connected. When discussing harmonic measure at infinity, we adjoin infinity to Ω and regard it as a domain on the Riemann sphere. Write μ for the equilibrium probability measure of K. We recall the following standard facts from logarithmic potential theory; see [5, 7] and [6, Theorems 1.9 and 1.12, Definition 3.4]. One has cap(H)=cap(K) (H)= (K), the equilibrium measures of H and K coincide, and suppμ⊂∂H⊂K μ⊂∂ H⊂ K. Frostman’s theorem and the exterior Green-function identity, with our sign convention, give Uμ(z) U_μ(z) =logcap(K) = (K) q.e. on H, .e. on H, (2.3) Uμ(w) U_μ(w) =logcap(K)+gΩ(w,∞) = (K)+g_ (w,∞) w∈Ω. w∈ . (2.4) Here gΩ(w,∞)>0g_ (w,∞)>0 for finite w∈Ωw∈ . The abbreviation “q.e.” means outside a set of logarithmic capacity zero. Such sets have planar area zero, so these identities also hold almost everywhere for m. For finite w∈Ωw∈ , let νw=ωΩ(w,⋅) _w= _ (w,·) be harmonic measure on ∂H∂ H. It is a probability measure, and harmonic measure at infinity is μ. Exterior logarithmic balayage gives Uνw(z)=log|z−w|−gΩ(w,∞)q.e. on H.U_ _w(z)= |z-w|-g_ (w,∞) .e. on H. (2.5) This formula requires no boundary smoothness and holds throughout H, including its interior. In the negative-log convention, [6, eqs. (3.15)–(3.16), p. 188–189] states the formula with the opposite sign for the additive constant. The identification with harmonic measure is also stated in [8, Section 2, eq. (2.4)]. For capacity one, (2.5) becomes Uνw(z)=log|z−w|−Uμ(w)q.e. on H.U_ _w(z)= |z-w|-U_μ(w) .e. on H. (2.6) We record explicitly the comparison needed to keep the subsequent perturbation positive. Lemma 2.1. Choose 0<r<R0<r<R with H⊂z:|z|<rH⊂\z:|z|<r\. For every |w|=R|w|=R, νw≤Cr,Rμ,Cr,R=R+rR−r. _w≤ C_r,Rμ, C_r,R= R+rR-r. (2.7) In particular, R=2rR=2r gives νw≤3μ _w≤ 3μ. Proof. For a Borel set A⊂∂HA⊂∂ H, the function uA(w)=νw(A)u_A(w)= _w(A) is bounded, nonnegative, and harmonic in Ω , with value μ(A)μ(A) at infinity. Thus vA(ζ)=uA(r/ζ)v_A(ζ)=u_A(r/ζ) extends harmonically across ζ=0ζ=0 to the unit disk, with vA(0)=μ(A)v_A(0)=μ(A). The disk Harnack inequality gives vA(ζ)≤1+|ζ|1−|ζ|vA(0).v_A(ζ)≤ 1+|ζ|1-|ζ|\,v_A(0). Apply it with ζ=r/wζ=r/w and |w|=R|w|=R. The resulting inequality holds for every Borel A, which is exactly the measure inequality (2.7). The zero function is included by continuity, or by applying Harnack to vA+ϵv_A+ε and letting ϵ↓0ε 0. ∎ 3 Harmonic polynomials positive outside a thin corridor The next lemma is independent of potential theory. Its probability measure need not be an equilibrium measure. Lemma 3.1. Let H⊂ℂH be compact and full, with m(H)>0m(H)>0, and let μ be a probability measure supported on H. Given η>0η>0, there exist a nonempty compact set E⊂HE⊂ H and a complex polynomial P such that m(H∖E)<η,∫Pμ=1,supE|P|<12.m(H E)<η, P\,dμ=1, _E|P|< 12. (3.1) Consequently h=Re(1−P)h= (1-P) is a real harmonic polynomial satisfying ∫hμ=0,h>12 on E. h\,dμ=0, h> 12 on E. (3.2) Proof. Consider the Cauchy transform Cμ(w)=∫dμ(z)w−z,w∈ℂ∖H.C_μ(w)= dμ(z)w-z, w H. It is holomorphic there and satisfies Cμ(w)=w−1+O(w−2)C_μ(w)=w^-1+O(w^-2) near infinity. Fix w0w_0 outside a large disk containing H, and consider the Taylor series of CμC_μ at w0w_0. Its radius of convergence R0R_0 is finite and positive. Otherwise its sum would be an entire function agreeing with CμC_μ on the connected set ℂ∖HC H, by the identity theorem. This entire function would tend to zero at infinity and hence would be identically zero by Liouville’s theorem, contradicting the coefficient of w−1w^-1. Let T denote the Taylor sum on the disk D=B(w0,R0)D=B(w_0,R_0). There is a point a∈∂Da∈∂ D at which T does not extend holomorphically across the boundary. This is the usual boundary-singularity property of a Taylor disk with finite radius of convergence: extensions at every boundary point would, by a finite cover of the boundary circle, extend the Taylor sum to a larger disk. Set γ=[w0,a]γ=[w_0,a]. For δ>0δ>0, let Vδ=z:dist(z,γ)<δ,E=H∖Vδ.V_δ=\z: (z,γ)<δ\, E=H V_δ. Since a line segment has planar area zero, choose δ>0δ>0 such that m(H∖E)<minη,m(H)/2.m(H E)< \η,m(H)/2\. Then E is compact and has positive area. Define the complex-linear functional Λ(Q)=∫Qμ (Q)= Q\,dμ on complex polynomials. We claim that there is no finite M such that |Λ(Q)|≤MsupE|Q|for every polynomial Q.| (Q)|≤ M _E|Q| every polynomial Q. (3.3) Suppose otherwise. Then Λ is a bounded functional on the polynomial restrictions in C(E)C(E). Complex Hahn–Banach and the Riesz representation theorem give a finite complex Borel measure λ supported on E such that ∫Qλ=∫Qμfor all complex polynomials Q. Q\,dλ= Q\,dμ all complex polynomials Q. Their analytic moments coincide, so their Cauchy transforms have the same Laurent expansion near infinity. Hence Cλ=CμC_λ=C_μ near infinity and, by the identity theorem, throughout ℂ∖HC H. In particular they agree near w0w_0. Since E is disjoint from VδV_δ, the function CλC_λ is holomorphic on VδV_δ. Both D and VδV_δ are convex, so D∩VδD∩ V_δ is connected. The identity theorem now gives Cλ=TC_λ=T on this intersection. But VδV_δ contains an open neighborhood of a. Thus CλC_λ extends T across a, a contradiction. This proves the claim. Choose Q for which |Λ(Q)|>2supE|Q|| (Q)|>2 _E|Q| and set P=Q/Λ(Q)P=Q/ (Q). This gives (3.1), and (3.2) follows by taking real parts. ∎ Remark 3.2. The removed corridor is allowed to pass through H. There is no assertion that a∈Ha∈ H, and the proof does not require E to be full or polynomially convex. These distinctions avoid making any assumption about continuation of CμC_μ through the original support. 4 Realization by a controlled signed measure Return to a compact K with cap(K)=1 (K)=1, its hull H, and its equilibrium measure μ. Lemma 4.1. Let h be a real harmonic polynomial with ∫hμ=0 h\,dμ=0. There is a finite real signed measure σ supported on K such that σ(K)=0,|σ|≤Chμ,Uσ=hm-a.e. on Hσ(K)=0, |σ|≤ C_hμ, U_σ=h m-a.e. on H (4.1) for some finite constant ChC_h. Proof. Write mk=∫zkμ(z)m_k= z^k\,dμ(z). The centered condition allows us to write h(z)=Re∑k=1dak(zk−mk).h(z)= _k=1^da_k(z^k-m_k). (4.2) If h=0h=0, take σ=0σ=0. Otherwise choose R larger than the moduli of all points of H, and put wθ=Reiθw_θ=Re^iθ. The uniformly convergent logarithmic expansion and (2.6) give, for each θ, Uνwθ(z)=−Re∑j=1∞zj−mjjRje−ijθm-a.e. on H.U_ _w_θ(z)=- _j=1^∞ z^j-m_jjR^je^-ijθ m-a.e. on H. (4.3) The series on the right converges uniformly for (z,θ)∈H×[0,2π](z,θ)∈ H×[0,2π]. Define the real trigonometric polynomial b(θ)=−2∑k=1dkRkRe(akeikθ),b(θ)=-2 _k=1^dkR^k (a_ke^ikθ), (4.4) and the real signed measure σ(A)=∫02πb(θ)νwθ(A)dθ2π.σ(A)= _0^2πb(θ) _w_θ(A) dθ2π. (4.5) Harmonic measure is a Borel probability kernel in its pole, so this integral defines a countably additive finite signed measure. For example, countable additivity follows from dominated convergence applied to disjoint unions, since b is integrable and each νwθ _w_θ is a probability. Its total mass is zero because b has zero mean. Its support lies in ∂H⊂K∂ H⊂ K. By Lemma 2.1, for a constant CRC_R independent of θ, one has νwθ≤CRμ _w_θ≤ C_Rμ. Therefore |σ|(A)≤CR(∫02π|b(θ)|dθ2π)μ(A).|σ|(A)≤ C_R ( _0^2π|b(θ)| dθ2π )μ(A). (4.6) To see the total-variation inequality, first bound |σ(Aj)||σ(A_j)| for each set in a finite Borel partition of A, sum, and then take the supremum over such partitions. We may integrate (4.3) against b(θ)dθ/(2π)b(θ)dθ/(2π) and interchange the logarithmic integrations. Indeed, for any bounded disk B containing H, ∫02π|b(θ)|∫B∫|log|z−t||dνwθ(t)m(z)dθ2π<∞ _0^2π\!|b(θ)| _B\! | |z-t||\,d _w_θ(t)\,dm(z) dθ2π<∞ by domination and (2.2). Fubini also deals with the exceptional sets in (4.3): they may depend on θ, but each has planar area zero. No common exceptional set for all poles is required. For j,k≥1j,k≥ 1, Fourier orthogonality gives ∫02πRe(akeikθ)Re(vje−ijθ)dθ2π=12Re(akvk),j=k,0,j≠k. _0^2π (a_ke^ikθ) (v_je^-ijθ) dθ2π= cases 12 (a_kv_k),&j=k,\\ 0,&j≠ k. cases Equations (4.3)–(4.5) consequently yield Uσ=hU_σ=h almost everywhere on H. Together with (4.6), this proves (4.1). ∎ 5 From probability measures to monic polynomials The next lemma uses planar area, rather than a measure on the boundary. It is valid for every compact set, including sets with irregular or positive-area boundary. Lemma 5.1. Let K⊂ℂK be compact, and let ρj,ρ∈(K) _j,ρ (K) be probability measures with ρj→ρ _j→ρ weakly. For every bounded disk B⊂ℂB , ‖Uρj−Uρ‖L1(B)⟶0.\|U_ _j-U_ρ\|_L^1(B) 0. Proof. For L>0L>0, set kL(z,t)=maxlog|z−t|,−Lk_L(z,t)= \ |z-t|,-L\, taking the value −L-L at z=tz=t. For fixed L, this is continuous and bounded on B¯×K B× K. Weak convergence gives pointwise convergence of the corresponding truncated potentials, and bounded convergence gives their convergence in L1(B)L^1(B). For any probability measure τ on K, Tonelli’s theorem gives the uniform truncation estimate ∫B∫(kL(z,t)−log|z−t|)τ(t)m(z) _B\! (k_L(z,t)- |z-t| )\,dτ(t)\,dm(z) ≤∫|u|<e−L(−L−log|u|)m(u)=π2e−2L. 28.45274pt≤ _|u|<e^-L(-L- |u|)\,dm(u)= π2e^-2L. (5.1) The triangle inequality, followed first by j→∞j→∞ and then by L→∞L→∞, proves the assertion. ∎ Proposition 5.2. For compact K⊂ℂK and ρ∈(K)ρ (K), ϑ(K)≤mz∈ℂ:Uρ(z)≤0. (K)≤ m\z :U_ρ(z)≤ 0\. (5.2) Proof. Choose equally weighted empirical probability measures ρN=1N∑j=1Nδzj,N,zj,N∈K, _N= 1N _j=1^N _z_j,N, z_j,N∈ K, converging weakly to ρ, with N→∞N→∞ along a sequence. For completeness, partition K into finitely many Borel sets of diameter tending to zero, approximate their masses by nonnegative rational numbers with a common denominator and total mass one, and place the prescribed multiplicities at a point of each nonempty partition cell. Taking the total error in the masses to zero produces the stated approximation. Increasing denominators if necessary makes N→∞N→∞. Set pN(z)=∏j=1N(z−zj,N)p_N(z)= _j=1^N(z-z_j,N). Then N−1log|pN|=UρN^-1 |p_N|=U_ _N, with the usual value −∞-∞ at zeros. If K⊂|t|≤rK⊂\|t|≤ r\, every probability measure τ on K satisfies Uτ(z)>0(|z|>r+1).U_τ(z)>0 (|z|>r+1). (5.3) Choose one bounded disk B containing |z|≤r+1\|z|≤ r+1\. It contains all the unit lemniscates under consideration. For s>0s>0, the part of UρN<0\U_ _N<0\ on which Uρ>sU_ρ>s has area at most s−1‖UρN−Uρ‖L1(B)s^-1\|U_ _N-U_ρ\|_L^1(B). Thus mUρN<0≤m(B∩Uρ≤s)+s−1∥UρN−Uρ∥L1(B).m\U_ _N<0\≤ m (B∩\U_ρ≤ s\ )+s^-1\|U_ _N-U_ρ\|_L^1(B). By Lemma 5.1, first letting N→∞N→∞ and then s↓0s 0 yields lim supNm|pN|<1≤m(B∩Uρ≤0)=mUρ≤0. _Nm\|p_N|<1\≤ m (B∩\U_ρ≤ 0\ )=m\U_ρ≤ 0\. This proves (5.2). ∎ Remark 5.3. The limiting set in (5.2) is the nonpositive sublevel set. No assertion about the area of the zero level set is needed. This distinction is useful because the equilibrium potential at capacity one can vanish on a set of positive area. 6 Proof of the capacity-one theorem Proof of Theorem 1.1. Let H=K^H= K and let μ be the equilibrium measure of K. If m(H)=0m(H)=0, then Uμ>0U_μ>0 outside H, so mUμ≤0=0m\U_μ≤ 0\=0. Proposition 5.2 gives ϑ(K)=0 (K)=0 immediately. Suppose m(H)>0m(H)>0 and fix η>0η>0. Lemma 3.1 gives E⊂HE⊂ H and a real harmonic polynomial h such that m(H∖E)<η,∫hμ=0,h>12 on E.m(H E)<η, h\,dμ=0, h> 12 on E. By Lemma 4.1, choose σ with σ(K)=0σ(K)=0, |σ|≤Chμ|σ|≤ C_hμ, and Uσ=hU_σ=h almost everywhere on H. For 0<t<[2(1+Ch)]−10<t<[2(1+C_h)]^-1, define ρt=μ+tσ. _t=μ+tσ. (6.1) The measure ρt _t is a probability supported on K: its density relative to μ is at least 1−tCh>1/21-tC_h>1/2, and its mass is one. Since Uμ=0U_μ=0 almost everywhere on H, Uρt=th>t/2m-a.e. on E.U_ _t=th>t/2 m-a.e. on E. Consequently m(H∩Uρt≤0)<η.m (H∩\U_ _t≤ 0\ )<η. (6.2) For each finite z∈Ω=ℂ∖Hz∈ =C H, the potential Uσ(z)U_σ(z) is finite and Uμ(z)=gΩ(z,∞)>0U_μ(z)=g_ (z,∞)>0. Hence Uρt(z)=Uμ(z)+tUσ(z)⟶Uμ(z)>0(t↓0).U_ _t(z)=U_μ(z)+tU_σ(z) U_μ(z)>0 (t 0). Every nonpositive sublevel set in this family lies in one bounded disk by (5.3). Dominated convergence applied to its intersection with Ω therefore gives m(Ω∩Uρt≤0)⟶0(t↓0).m ( ∩\U_ _t≤ 0\ ) 0 (t 0). (6.3) Choose t small enough that the left-hand side is less than η. Combining (6.2) and (6.3), we obtain mUρt≤0<2ηm\U_ _t≤ 0\<2η. Proposition 5.2 now gives ϑ(K)<2η (K)<2η. Since η>0η>0 was arbitrary and ϑ(K)≥0 (K)≥ 0, the theorem follows. Equivalently, given a>0a>0, first take 2η<a2η<a and then choose an empirical approximation far enough along the sequence to obtain a polynomial with unit-lemniscate area less than a. ∎ 7 Completion of the vanishing question For completeness, we first recover the qualitative capacity-greater-than-one theorem of Ghosh and Ramachandran [3, Theorem 3.1] from Proposition 5.2. Proposition 7.1. If K⊂ℂK is compact and cap(K)>1 (K)>1, then ϑ(K)=0 (K)=0. Proof. The equilibrium potential equals logcap(K)>0 (K)>0 quasi-everywhere on its hull and equals this constant plus the positive exterior Green function outside the hull. Its nonpositive set therefore has planar area zero. Apply Proposition 5.2. ∎ The sharp exponential decay rate established in [3, Theorem 3.3] is a stronger conclusion; the argument here recovers only the vanishing of the infimum. Proposition 7.2. If F⊂ℂF is unbounded, then ϑ(F)=0 (F)=0. Proof. Choose a,b∈Fa,b∈ F at distance D>2D>2 and set p(z)=(z−a)(z−b)p(z)=(z-a)(z-b). If |p(z)|<1|p(z)|<1, the smaller of |z−a||z-a| and |z−b||z-b| is less than one. The larger is therefore greater than D−1D-1, so the smaller is less than (D−1)−1(D-1)^-1. Thus |p|<1⊂B(a,(D−1)−1)∪B(b,(D−1)−1),\|p|<1\⊂ B (a,(D-1)^-1 )∪ B (b,(D-1)^-1 ), and m|p|<1≤2π(D−1)−2m\|p|<1\≤ 2π(D-1)^-2. Let D→∞D→∞. ∎ Corollary 7.3. For every closed infinite set F⊂ℂF of transfinite diameter at least one, ϑ(F)=0 (F)=0. Proof. If F is bounded, it is compact; apply Theorem 1.1 or Proposition 7.1. If it is unbounded, use Proposition 7.2. ∎ The unbounded argument does not assume the existence of a compact subset of capacity greater than one. Such an assumption would fail for an unbounded closed discrete set. 8 Further remarks Theorem 1.1 removes the boundary regularity assumption from the capacity-one theorem of [4, Theorem 6]. The general perturbation strategy comes from that work, while Sections 3–5 provide the arguments needed for arbitrary compact sets. In particular, the proof controls planar area through L1L^1 convergence of potentials; it does not require uniform convergence at their logarithmic singularities. The order of choices is essential: first the corridor and harmonic polynomial, then the signed measure, then the positive perturbation size, and finally the empirical approximation. None of the intermediate constants is asserted to be uniform as the corridor narrows. Obtaining degree bounds or explicit root configurations is a separate question. Thus Corollary 7.3 gives the affirmative answer to the vanishing question in [1, Problem 4], while the separate question of determination by transfinite diameter alone has the negative answer given by the counterexamples in [2, Section 3.3] and [3, Example 2.1]. Appendix A A normalization check on the unit circle Let K=z:|z|=1K=\z:|z|=1\, so H=¯H= D and dμ(eiϕ)=dϕ/(2π)dμ(e^iφ)=dφ/(2π). For the centered harmonic polynomial h(z)=Rezh(z)= z, define dσ(eiϕ)=−2cosϕdϕ2π.dσ(e^iφ)=-2 φ\, dφ2π. Then σ(K)=0σ(K)=0 and |σ|≤2μ|σ|≤ 2μ. For |z|<1|z|<1, expand the logarithm and integrate term by term: ∫log|z−eiϕ|dσ(eiϕ)=∫(−Re∑k≥1zke−ikϕk)(−2cosϕ)dϕ2π=Rez. |z-e^iφ|\,dσ(e^iφ)= (- _k≥ 1 z^ke^-ikφk )(-2 φ) dφ2π= z. For |z|>1|z|>1, the exterior expansion similarly gives Uσ(z)=Re(1/z)U_σ(z)= (1/z). Continuity of the logarithmic single-layer potential with this smooth density extends the interior equality to |z|=1|z|=1. In particular, for 0<t<1/20<t<1/2, the measure dρt(eiϕ)=(1−2tcosϕ)dϕ2πd _t(e^iφ)=(1-2t φ) dφ2π is positive and has potential tRezt z on the closed disk. This example checks the sign and factor in the representation lemma; by itself it does not give arbitrarily small area. That step requires the harmonic polynomials furnished by Lemma 3.1. AI assistance and review status The AI-assisted proposed result was obtained through the author’s prompting strategy using OpenAI’s GPT-6-Astra model through Codex. AI assistance included proof exploration, mathematical reasoning, literature checks, manuscript drafting, and internal reviews by parallel reasoning agents. These internal checks are not independent human verification. No proof-assistant formalization or external mathematical endorsement is asserted in this draft. The bibliography distinguishes the existing results and method from the proposed extension. References [1] P. Erdős, F. Herzog, and G. Piranian (1958) Metric properties of polynomials. Journal d’Analyse Mathématique 6, p. 125–148. External Links: Document, Link Cited by: §1, §1, §8. [2] T. Feng, T. Trinh, G. Bingham, J. Kang, S. Zhang, S. Kim, K. Barreto, C. Schildkraut, J. Jung, J. Seo, C. Pagano, Y. Chervonyi, D. Hwang, K. Hou, S. Gukov, C. Tsai, H. Choi, Y. Jin, W. Li, H. Wu, R. Shiu, Y. Shih, Q. V. Le, and T. Luong (2026) Semi-autonomous mathematics discovery with Gemini: a case study on the Erdős problems. Note: Version 1; see Section 3.3 External Links: 2601.22401, Link Cited by: §1, §8. [3] S. Ghosh and K. Ramachandran (2026) A note on the Erdős minimal area problem. Proceedings of the American Mathematical Society. Note: Accepted for publication; cited preprint version 3, 27 August 2026 External Links: Document, 2604.03036, Link Cited by: §1, §1, §7, §7, §8. [4] M. Krishnapur, E. Lundberg, and K. Ramachandran (2025) On the area of polynomial lemniscates. Note: Version 1, 24 March 2025 External Links: 2503.18270, Document, Link Cited by: §1, §1, §8. [5] T. Ransford (1995) Potential theory in the complex plane. London Mathematical Society Student Texts, Vol. 28, Cambridge University Press. External Links: ISBN 9780521461207, Document, Link Cited by: §2. [6] E. B. Saff (2010) Logarithmic potential theory with applications to approximation theory. Surveys in Approximation Theory 5, p. 165–200. External Links: 1010.3760, Link Cited by: §2, §2. [7] E. B. Saff and V. Totik (1997) Logarithmic potentials with external fields. 1 edition, Grundlehren der mathematischen Wissenschaften, Vol. 316, Springer-Verlag, Berlin, Heidelberg. External Links: Document, Link Cited by: §2. [8] F. Toókos (2005) Smoothness of Green’s functions and density of sets. Acta Scientiarum Mathematicarum (Szeged) 71 (1–2), p. 117–146. External Links: Link Cited by: §2.