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ERDőS #1040 · PARTIAL

Erdős problem 1040 — live audit, exact frontier, and a sharp Chebyshev computation

Access date: 2026-07-31 UTC.

Claim labels used throughout:

0. Mandatory live-page audit

I fetched https://www.erdosproblems.com/1040, its LaTeX view, and its discussion thread through the Bright Data browser route, not datacenter curl.

The following is the verbatim statement from the live LaTeX view:

Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of\[\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,\]as $f$ ranges over all polynomials of the shape $\prod (z-z_i)$ with $z_i\in F$.

Is $\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\mu(F)=0$ whenever the transfinite diameter of $F$ is $\geq 1$?

Live status and collision check:

Thus the prescribed stop condition does not fire. The comment is a literature pointer, not a claimed-proof marker, and the cited paper itself explicitly says that its answer to the second question is partial.

The known results displayed on the live page are:

  1. Erdős–Herzog–Piranian (1958) prove the desired conclusion for a line segment or circular disk, and prove that when the transfinite diameter is below \(1\), every lemniscate contains a disk of radius bounded below in terms of \(F\).
  2. Erdős–Netanyahu (1973) make the latter radius depend only on the capacity \(c\in(0,1)\) when \(F\) is bounded, closed, and connected.
  3. The page defines the transfinite diameter by the Vandermonde limit shown in the statement’s notes.
  4. Feng et al. (2026), reporting Aletheia’s construction, disprove determination of \(\mu(F)\) by capacity using two countable compact capacity-zero sets.

These page facts are source-audited rather than new mathematical claims.

1. Primary-source literature audit

I checked the following primary documents, not only search snippets.

| Source | What the document actually says | Verification | |---|---|---| | Erdős–Herzog–Piranian, Metric properties of polynomials (1958), p. 135, Problem 4 | Gives the original problem in essentially the live wording; Theorem 6 handles transfinite diameter \(<1\); the paragraph after Problem 4 names line segments and circular disks. | PDF SHA-256 5d20ad770563e425ac3343e31581859a24facabf744f10489b8f8d201b0ec12a | | Erdős–Netanyahu, A remark on polynomials and the transfinite diameter (1973), pp. 23–25 | Its theorem gives a disk of radius \(\rho(c)>0\) for bounded closed connected sets of capacity \(1-c\). The final page explicitly says the general capacity-one case is open. | PDF SHA-256 afda8742064294060884388b21758d3db09f1ba914aeb8d57bc514332bb88a59 | | Feng et al., arXiv:2601.22401, §3.2 | Gives the capacity-zero counterexample. Remark 3.2 explicitly says Aletheia’s attempted second part was incorrect and was omitted. | PDF SHA-256 fa2afcd192e6209183f44840f11632b266f2b973b722bb0b346ee39e3edee76e | | Krishnapur–Lundberg–Ramachandran, arXiv:2503.18270, Theorem 6 and Proposition 14 | Theorem 6 proves \(\inf_n\kappa_n(K,1)=0\) when \(K\) is the closure of a bounded \(C^2\)-smooth domain of capacity one. Proposition 14 handles compact unit polynomial lemniscates quantitatively. It does not prove arbitrary capacity-one compact sets. | PDF SHA-256 49ada70f29f7fa035a2c993278f6d1e6df2ce02543ff02d02b6ff24d50f607f0 | | Ghosh–Ramachandran, arXiv:2604.03036v2, Theorem 3.1, Theorem 4.1, Lemma 4.6 | For every compact \(K\) with \(\operatorname{cap}(K)>1\), Fekete polynomials have exponentially shrinking area. It gives an \(O(1/n)\) Chebyshev construction for \([-2,2]\), and handles period-\(d\) capacity-one sets. The introduction says arbitrary capacity-one compact sets remain open. | PDF SHA-256 2646cf798aec63e9f97aa8b1585bdc449acafce9c09a077577072b29d3e8c07e |

Targeted arXiv/web searches through the access date found no later primary paper claiming the arbitrary capacity-one case. This is a search miss report, not a claim of exhaustive bibliographic completeness.

The current literature state is therefore:

2. A clean reduction for the live statement

The modern papers formulate the problem for compact \(K\), whereas the live statement permits any closed infinite \(F\). The following elementary observation closes that gap for unbounded sets.

Proposition 2.1: every unbounded \(F\) has \(\mu(F)=0\) (a)

Choose \(a,b\in F\), and put \(d=|a-b|>2\). Translation and rotation preserve area and polynomial modulus, so it suffices to study

\[ p_d(z)=(z-d/2)(z+d/2)=z^2-d^2/4. \]

Set \(A=d^2/4>1\) and \(v=z^2\). The two inverse branches of \(z\mapsto z^2\) have total area Jacobian

\[ \sum_{z^2=v}\left|\frac{dz}{dv}\right|^2 =2\frac1{4|v|}=\frac1{2|v|}. \]

Consequently,

\[ \begin{aligned} \left|\{z:|p_d(z)|<1\}\right| &=\int_{|v-A|<1}\frac{dA(v)}{2|v|}\\ &\leq \frac{\pi}{2(A-1)} =\frac{2\pi}{d^2-4}. \end{aligned} \]

An unbounded \(F\) contains pairs with \(d\to\infty\), so the displayed upper bound tends to zero. Hence \(\mu(F)=0\).

Corollary 2.2: exact remaining regime (b)

A bounded closed subset of \(\mathbb C\) is compact. Combining Proposition 2.1 with Ghosh–Ramachandran Theorem 3.1 gives:

\[ \boxed{\text{The only unresolved part of the second question is compact }F \text{ with }\operatorname{cap}(F)=1.} \]

This is rigorous modulo Ghosh–Ramachandran Theorem 3.1. It also prevents an accidental overstatement: their theorem has the strict hypothesis \(\operatorname{cap}(F)>1\), not \(\geq1\).

3. New exact computation for the threshold Chebyshev construction

Ghosh–Ramachandran prove an \(81\pi/(2n)\) upper bound for the lemniscate of the monic Chebyshev polynomial on \([-2,2]\). The following calculation gives its exact area integral and sharp leading constant. I did not locate this formula in the primary sources searched above; no novelty beyond that search is asserted.

Define

\[ \mathcal T_n(z)=2T_n(z/2),\qquad L_n=\left|\{z:|\mathcal T_n(z)|<1\}\right|. \]

Here \(\mathcal T_n\) is monic of degree \(n\), and every zero lies in the capacity-one set \([-2,2]\).

Theorem 3.1: exact area formula and sharp asymptotic (a)

For every \(n\geq3\), let

\[ \phi(w)=\arcsin(w/2) \]

be the branch analytic on \(|w|<1\). Then

\[ \boxed{ L_n=\frac2n\int_{|w|<1} \frac{\cosh\!\left(\frac{2\,\operatorname{Im}\phi(w)}n\right)} {|4-w^2|}\,dA(w). } \tag{3.1} \]

Put

\[ \begin{aligned} C_*&=2\int_{|w|<1}\frac{dA(w)}{|4-w^2|}\\ &=\frac{\pi}{2}\sum_{m=0}^{\infty} \frac{\binom{2m}{m}^2}{256^m(2m+1)}\\ &=\frac{\pi}{2}\, {}_3F_2\!\left(\frac12,\frac12,\frac12;1,\frac32;\frac1{16}\right)\\ &=1.5791556883541794184942310069743094\ldots . \end{aligned} \tag{3.2} \]

Writing \(M=\operatorname{arsinh}(1/2)\), one has the explicit sharp bracket

\[ \boxed{ \frac{C_*}{n}\leq L_n \leq \frac{C_*}{n}\cosh\!\left(\frac{2M}{n}\right). } \tag{3.3} \]

In particular,

\[ L_n=\frac{C_*}{n}+O(n^{-3}) \quad\text{and}\quad \lim_{n\to\infty}nL_n=C_*. \tag{3.4} \]

This is an explicit admissible construction, so it also proves

\[ A_n([-2,2])\leq L_n \leq \frac{C_*}{n}\cosh\!\left(\frac{2M}{n}\right), \]

where \(A_n(K)\) is the degree-\(n\) constrained minimum. It does not assert that the Chebyshev polynomial minimizes area.

Proof of Theorem 3.1

The critical points of \(\mathcal T_n\) are \(2\cos(j\pi/n)\), \(1\leq j<n\), and their critical values are \(2(-1)^j\). Thus there is no critical value in the unit disk. The \(n\) inverse branches over that disk are

\[ z_k(w)=2\cos\!\left(\theta_k+\frac{(-1)^k}{n}\phi(w)\right), \qquad \theta_k=\frac{(2k-1)\pi}{2n}, \quad 1\leq k\leq n. \tag{3.5} \]

Indeed,

\[ 2\cos\!\left(n\theta_k+(-1)^k\phi(w)\right)=w. \]

The change-of-variables formula on the \(n\) components gives

\[ L_n=\int_{|w|<1}\sum_{k=1}^n|z_k'(w)|^2\,dA(w). \tag{3.6} \]

Write \(\phi(w)=a+ib\). Since

\[ \left|\sin(x+iy)\right|^2 =\frac{\cosh(2y)-\cos(2x)}2, \]

differentiating (3.5) yields

\[ \begin{aligned} \sum_{k=1}^n|z_k'(w)|^2 =\frac{4|\phi'(w)|^2}{n^2} \bigg[ \frac n2\cosh(2b/n) -\frac12\sum_{k=1}^n \cos\!\left(\frac{(2k-1)\pi}{n} +(-1)^k\frac{2a}{n}\right) \bigg]. \end{aligned} \tag{3.7} \]

The cosine sum vanishes for every real \(a\) when \(n\geq3\). For even \(n\geq4\), the even- and odd-index exponential sums are separate finite geometric sums with ratio \(e^{4\pi i/n}\), hence both vanish. For odd \(n\), pair \(k\) with \(n+1-k\); the pair contribution is \(2\cos((2k-1)\pi/n)\cos(2a/n)\), and the middle term completes the real-part sum of all \(n\)-th roots of \(-1\), which is zero.

Finally,

\[ \phi'(w)=\frac1{\sqrt{4-w^2}}, \qquad |\phi'(w)|^2=\frac1{|4-w^2|}. \]

Substitution into (3.6) proves (3.1).

For (3.2), set

\[ h(w)=(1-w^2/4)^{-1/2} =\sum_{m\geq0}\frac{\binom{2m}{m}}{16^m}w^{2m}. \]

Orthogonality of monomials in the disk gives

\[ \int_{|w|<1}\frac{dA(w)}{|4-w^2|} =\frac14\int_{|w|<1}|h(w)|^2\,dA(w) =\frac{\pi}{4}\sum_{m\geq0} \frac{\binom{2m}{m}^2}{256^m(2m+1)}. \]

This proves the series and hypergeometric forms.

If \(u=\phi(w)=x+iy\), then \(|\sin u|=|w|/2<1/2\), while

\[ |\sin(x+iy)|^2=\sin^2x+\sinh^2y. \]

Therefore \(|y|<M=\operatorname{arsinh}(1/2)\). Bounding the cosh factor in (3.1) between \(1\) and \(\cosh(2M/n)\) proves (3.3), and its Taylor expansion proves (3.4). \(\square\)

4. Recomputed values

The following values are (d) computational-only evaluations of the proved formula (3.1). They are not assumed in its proof.

| \(n\) | \(L_n\) | \(nL_n\) | |---:|---:|---:| | 3 | 0.533169528806692 | 1.599508586420077 | | 4 | 0.397645690102206 | 1.590582760408825 | | 5 | 0.317292534813828 | 1.586462674069138 | | 6 | 0.264037932301169 | 1.584227593807016 | | 8 | 0.197750912370118 | 1.582007298960947 | | 10 | 0.158098032345447 | 1.580980323454466 | | 20 | 0.078980585763331 | 1.579611715266619 | | 50 | 0.031584572935077 | 1.579228646753825 | | 100 | 0.015791739277432 | 1.579173927743167 |

5. Independent checker

Standalone checker:

runs/erdos1040_wavew050_verify.py

Run from the repository root with:

python runs/erdos1040_wavew050_verify.py

It performs four independent groups of checks:

  1. constructs \(\mathcal T_n\) from the recurrence and checks monicity, roots, critical points/values, and every inverse-branch identity at deterministic complex samples;
  2. checks the trigonometric cancellation separately;
  3. evaluates (3.1) by tensor Gauss–Legendre disk quadrature and compares it with a Green/shoelace area computed from all \(n\) boundary branches, without using the collapsed area integrand;
  4. recomputes \(C_*\) from both the positive binomial series and disk quadrature, checks (3.3), and checks the far-separated quadratic estimate.

Observed run time was 6.6 seconds. The final output was:

algebra PASS: roots 7.367e-13, critical derivatives 5.765e-12, critical values 3.517e-13, inverse branches 3.128e-14
trigonometric cancellation PASS: max residual 1.166e-14
limiting constant PASS: series=1.579155688354179, disk=1.579155688354189, series tail < 1.12e-52
area formula PASS: max polygon relative discrepancy 4.130e-10; max sharp-bound violation 0.000e+00
far-separated-root bound PASS
ALL CHECKS PASS

The positive series tail estimate uses the elementary ratio bound \(a_{m+1}/a_m<1/16\); after 40 terms its contribution to \(C_*\) is below \(1.12\times10^{-52}\). The displayed decimal quadratures remain class (d).

6. Exact wall for the general capacity-one case

Ghosh–Ramachandran’s \(>1\) proof uses Fekete points to obtain

\[ |p_n'(z_{j,n})|\geq \operatorname{cap}(K)^{n-1}. \]

At capacity one this degenerates to the non-decaying bound \(1\), so their Bernstein-diameter argument gives no shrinking components. This is not merely a loose estimate: for the unit disk, the Fekete polynomials are \(z^n-1\), whose unit lemniscate areas stay bounded away from zero even though \(\mu(\overline{\mathbb D})=0\). (b: Ghosh–Ramachandran, introduction and proof of Theorem 3.1)

Their univalent-function Lemma 4.3 isolates a precise sufficient replacement. Suppose \(p\) has simple zeros \(z_1,\ldots,z_n\in K\), and for some fixed \(r>1\) every component of \(\{|p|<r\}\) contains exactly one zero. Then (b: Ghosh–Ramachandran Lemma 4.3)

\[ \left|\{|p|<1\}\right| \leq \frac{\pi r^4}{(r-1)^4} \sum_{j=1}^n\frac1{|p'(z_j)|^2}. \tag{6.1} \]

Thus this machinery would settle arbitrary compact capacity-one \(K\) if one could prove the following missing lemma:

For every compact \(K\) of logarithmic capacity one, construct monic \(p_m\) with all zeros in \(K\), a fixed \(r>1\), one zero per component of \(\{|p_m|<r\}\), and \[ > \sum_{p_m(z)=0}|p_m'(z)|^{-2}\longrightarrow0. > \]

For \([-2,2]\), the Chebyshev sequence has critical values of modulus \(2\) and the reciprocal-derivative sum is exactly \(1/(2n)\), so (6.1) works. For disk Fekete polynomials, the derivative sum also decays, but the only critical value has modulus \(1\), so the required component separation fails. This identifies the threshold obstruction: derivative growth alone is insufficient; one needs a root sequence with simultaneous critical-value separation. (a) for the two explicit polynomial calculations; (b) for the criterion (6.1)

No finite computation over individual compact sets supplies the uniform existence statement in this missing lemma. Brute-force root selection from an \(M\)-point discretization costs \(\binom{M+n-1}{n}\) configurations before continuous refinement, so scaling such a search would be expensive and, more importantly, could not prove the required assertion for every compact capacity-one set. This is the honest wall rather than evidence against the conjecture.

PARTIAL: Reduced the live question exactly to compact capacity-one sets, proved an exact Chebyshev lemniscate-area formula with sharp constant \(C_*=1.579155688354179\ldots\), and isolated critical-value separation plus reciprocal-derivative decay as the missing uniform lemma.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger