Problem 1151

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claims/: The 1 claim page of Problem 1151, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $\mathcal{L}^nf(x)$.

Formulation. The site's commentary is unsure whether the Statement is meant at a fixed point. Erdős's sources read it at one. The booklet [Va99, 2.41] that the site follows recalls his divergence theorem at $x_0=\cos(\pi p/q)$ with $p\equiv q\equiv1\pmod 2$. His survey [Er67, p. 68] says that his 1943 corrections state, without proof, that at such a point every closed set is the set of limit points of $\mathcal{L}^nf(x_0)$ for some continuous $f$. This page reads the Statement so: at a fixed $x_0=\cos(\pi p/q)$ with $p,q$ odd, for every closed $A\subseteq[-1,1]$ as the site states, the empty set meaning $\lvert\mathcal{L}^nf(x_0)\rvert\to\infty$.

Status. 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, and the standing in the frontmatter follows from it.

Source. erdosproblems.com/1151, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1151, https://www.erdosproblems.com/1151.

References.

• [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).

Formalization. No formal-conjectures statement file exists for the problem, and the site's page reports no formalized statement. A Lean development of Theorem 1.1(a) of the claimed note is linked from the claim page; it is not built or audited in this repository.

Current assessment

The question as the Formulation reads it is OPEN on the site. Erdős's theorem [Er41], with the 1943 corrections, gives the case $A=\emptyset$ at the points $x_0=\cos(\pi p/q)$ with $p,q$ odd: there some continuous $f$ has $\lvert\mathcal{L}^nf(x_0)\rvert\to\infty$. One full claim is pending, the note recorded on [[problems/polynomials/E1151/claims/2026_04_30_chojecki|the claim page]]. Its Theorem 1.1 gives every nonempty closed $A$ at every fixed point, and the empty set exactly at the points $\cos(\pi\alpha)$ with $\alpha$ rational of odd denominator. By the note, the Statement read at an arbitrary fixed point fails only for $A=\emptyset$ away from those points. The note's Section 7 rules out the two other readings: the reading in which $A$ is the set of points with a nonempty cluster set, and the reading with one cluster set shared by every point. The claim is neither reviewed nor refereed, and its Lean development is not built in this repository. The derived standing is claimed, with claim value proved.

Search scope (2026-10-07): the site page; its proof-claims tab, which is empty; its discussion thread of seven comments; the community database at teorth/erdosproblems, which lists the problem as open and unformalized; the formal-conjectures tree; the note at ulam.ai; the Lean folder at its pinned commit.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1943_note_farey_series
• erdos_1941_divergence_properties_lagrange_interpolation_parabolas
• erdos_1941_divergence_properties_lagrange_interpolation_parabolas / theorem_1
• erdos_1941_divergence_properties_lagrange_interpolation_parabolas / theorem_2