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Updated Oct 8

Problem 1152

claims/: The 1 claim page of Problem 1152, one per claimant's result; the problem's standing derives from them.

Statement. For $n\geq 1$ fix some sequence of $n$ distinct numbers $x_{1n},\ldots,x_{nn}\in [-1,1]$ . Let $\epsilon=\epsilon(n)\to 0$ .

Does there always exist a continuous function $f:[-1,1]\to \mathbb{R}$ such that if $p_n$ is a sequence of polynomials, with degrees $\deg p_n<(1+\epsilon(n))n$ , such that $p_n(x_{kn})=f(x_{kn})$ for all $1\leq k\leq n$ , then $p_n(x)\not\to f(x)$ for almost all $x\in [-1,1]$ ?

Status. OPEN, the site's label (page last edited 23 January 2026; the problem page and its proof-claims tab accessed 2026-10-06). The tab carries one full proof claim, by Qiyuan Gu, submitted 2026-09-04 with a Zenodo write-up drafted, as the tab discloses, using GPT 6 Astra, running on top of GPT 5.6 Sol and Claude Fable 5.1; it claims to answer the question yes in a stronger form: for any array and any excess degree $r_n=o(n)$ some continuous $f$ makes every sequence of interpolants $p_n$ of degree at most $n+r_n$ satisfy $\limsup_n\lvert p_n(x)\rvert=\infty$ at almost every $x$ . The claim is recorded, unadopted, on its claim page; the derived standing departs from the site's label because this pending full claim makes the problem claimed as proved, and it stays pending since no outside review or refereed publication of it is known. For a fixed $\epsilon>0$ the opposite holds for suitable arrays: Erdős, Kroó and Szabados [EKS89] give arrays for which every continuous $f$ has interpolants of degree below $(1+\epsilon)n$ converging uniformly, as the site's commentary records.

Source. erdosproblems.com/1152, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1152, https://www.erdosproblems.com/1152.

References.

[EKS89] Erdős, P. and Kroó, A. and Szabados, J., On convergent interpolatory polynomials. J. Approx. Theory 58(2) (1989), 232-241.

Formalization. No formal-conjectures statement file exists for the problem, and the site's page reports no formalized statement. The claimant's Zenodo record carries a partial Lean 4 formalization that takes Remez's inequality and the remaining analytic estimates as hypotheses. It is linked from the claim page and is not built or audited in this repository.

Progress

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Known Results

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Linked library material

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Linked from (9)

Gu's almost-everywhere divergence with sublinear excess degree problems/polynomials/E1152/claims Polynomials polynomials/erdos_1989_convergent_interpolatory_polynomials Theorem: interpolants of degree [n(1+ε)] with error O(E_{[n(1+ε)]} f) exist exactly under a Chebyshev density bound and a spacing condition Theorem A: degree [dn(1+ε)] interpolants with error O(E_{[dn(1+ε)]} f) exist exactly when the density bound is d/π and the spacing condition holds polynomials/vertesi_2013_paul_erdos_interpolation_problems_results_new Theorem 4.1 (p. 724): uniformly bounded fundamental polynomials allow convergent interpolation of degree n(1+eps)Theorem 4.2 (p. 725): Erdős, Kroó and Szabados's node conditions for near-best interpolation of degree n(1+eps)

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