Problem 1151

Given $a_1,\ldots,a_n\in [-1,1]$ let

\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i)\ell_i(x)

be the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i$ for $1\leq i\leq n$ (that is, the Lagrange interpolation polynomial).

Let $a_i$ be the set of Chebyshev nodes. Prove that, for any closed $A\subseteq [-1,1]$ , there exists a continuous function $f$ such that $A$ is the set of limit points of $\mathcal{L}^nf(x)$ .

Source: erdosproblems.com/1151

Analysis
Polynomials

Wiki page

Status

ClaimedProved

A full solution has been claimed but not yet accepted. The statement is true.

The site labels the problem OPEN (page last edited 23 January 2026). Its discussion thread carries a note that Przemek Chojecki posted on 30 April 2026, produced with GPT-5.5 Pro as he wrote there and hosted at ulam.ai. The note claims the problem in full in the reading of the Formulation. The claim is recorded, unadopted, on its claim page (Chojecki, 2026), and the standing in the frontmatter follows from it.

Er41: Erdős, P., On divergence properties of the Lagrange interpolation parabolas. Ann. of Math. (2) 42 (1941), 309-315.
Er43: Erdős, P., A note on Farey series. Quart. J. Math. Oxford Ser. (1943), 82-85. The site's commentary cites under this key Erdős's statement, made without proof, that every closed set is the set of limit points at such a point. The site's reference record resolves the key to this note on Farey series, which contains nothing on interpolation. The statement is in P. Erdős, Corrections to two of my papers, Ann. of Math. (2) 44 (1943), 647-651, the paper that [Er67, p. 68] cites for it.
Er67: Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) 10 (33) (1968), 65-73.
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Problem 1151

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Proof claims

Discussion