Problem 1132

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claims/: The 1 claim page of Problem 1132, one per claimant's result; the problem's standing derives from them.

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Statement. For $x_1,\ldots,x_n\in [-1,1]$ let

$$l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$$

which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.

Let $x_1,x_2,\ldots\in [-1,1]$ be an infinite sequence, and let

$$L_n(x) = \sum_{1\leq k\leq n}\lvert l_k(x)\rvert,$$

where each $l_k(x)$ is defined above with respect to $x_1,\ldots,x_n$.

Must there exist $x\in (-1,1)$ such that

$$L_n(x) >\frac{2}{\pi}\log n-O(1)$$

for infinitely many $n$?

Is it true that

$$\limsup_{n\to \infty}\frac{L_n(x)}{\log n}\geq \frac{2}{\pi}$$

for almost all $x\in (-1,1)$?

Formulation. The Statement does not say whether the constant in the $O(1)$ term may depend on $x$ or on the sequence, a gap Tao [Ta26b, Remark 1.12] notes. Erdős's sources read it as one absolute constant. The booklet [Va99, 2.43] asks for the bound "with some absolute constant $c$", and [Er67, p. 68] writes an unadorned $c$, as for the absolute constants $c_1,\ldots,c_4$ of pp. 66-67. This page reads the first question so: is there an absolute $c$ such that every sequence has a point $x\in(-1,1)$ with $L_n(x)>\frac2\pi\log n-c$ for infinitely many $n$? Both sources pose the question for an arbitrary triangular array, of which the Statement's single sequence is a special case; for non-nested arrays Gu's companion note claims the answer no.

Status. The site labels the problem OPEN (page last edited 01 April 2026; proof-claims tab accessed 2026-10-07). The tab carries one proof claim, submitted as full, by Qiyuan Gu (using GPT-6 Astra, GPT-5.6 Sol, Claude Opus 5, as the tab writes it), posted 2026-09-05 with a Zenodo write-up: it claims the almost-everywhere $\limsup$ bound as asked, and the first question with a constant that may depend on the point $x$, while a companion note by the same claimant states that for a non-nested triangular array no constant uniform in the array can serve; for a single sequence, as the Statement is posed, the uniform reading is not settled. A comment on the claim objects that only the weaker, point-dependent variant is answered, and the Statement does not fix the dependence of the $O(1)$ term, a point Tao [Ta26b] had already raised. In the reading of the Formulation the claim would settle only the second question. It is recorded, unadopted, as a partial claim on its claim page, and the derived standing in the frontmatter is open. Tao [Ta26b] proves, for every $\omega(n)\to\infty$, a dense set of $x$ with $L_n(x)\ge\frac2\pi\log n-\omega(n)$ for infinitely many $n$, short of the constant the question asks for.

Source. erdosproblems.com/1132, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1132, https://www.erdosproblems.com/1132.

References.

• [Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degré n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.
• [Er61c] Erdős, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.
• [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.
• [Ta26b] T. Tao, Local Bernstein theory, and lower bounds for Lebesgue constants. arXiv:2603.21453 (2026).
• [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. The claimant's Zenodo record carries a Lean 4 archive said to formalize the claimed theorems, linked from the claim page; it is not built or audited in this repository.

Current assessment

The question (site formulation, page last edited 01 April 2026). For one infinite sequence of nodes in $[-1,1]$, with $L_n$ the Lebesgue function of its first $n$ terms: must some $x\in(-1,1)$ have $L_n(x)>\frac2\pi\log n-O(1)$ for infinitely many $n$, and is $\limsup L_n(x)/\log n\ge2/\pi$ for almost every $x$? OPEN. The first question is read, as the Formulation records, with one absolute constant. The rows of $L_n$ are nested, so the Statement is a special case of Erdős's question for an arbitrary triangular array in [Er67], and results for arrays transfer to it while counterexamples built from non-nested arrays do not.

Known results. The site's commentary records that a result of Bernstein [Be31] implies that the set of $x$ with $\limsup L_n(x)/\log n\ge2/\pi$ is everywhere dense, that Erdős [Er61c] proved $\max_{[-1,1]}L_n>\frac2\pi\log n-O(1)$ for every fixed row of nodes, and that Tao [Ta26b] (card) proved, for every $\omega(n)\to\infty$, a dense set of $x$ with $L_n(x)\ge\frac2\pi\log n-\omega(n)$ for infinitely many $n$, which falls short of a constant loss.

Pending claim. One partial claim, unadopted: Gu 2026, a Zenodo write-up submitted to the site's proof-claims tab on 2026-09-05, produced with GPT-6 Astra, GPT-5.6 Sol and Claude Opus 5 as the tab discloses. It states, for every triangular array, the almost-everywhere $\limsup$ bound and a dense set of points where $L_n(x)>\frac2\pi\log n-C(x)$ infinitely often with a point-dependent constant; a companion note states that for non-nested arrays no constant independent of the array can serve. It claims the second question. Whether one absolute constant serves for a single sequence, the first question as the Formulation reads it, is settled by neither document, and a comment on the claim objects that only the point-dependent variant is answered. The claim is neither reviewed nor refereed; the claimant's Lean 4 archive is not built or audited in this repository. The derived standing is open.

Search scope. The site's problem page and its proof-claims tab with its one claim and one comment; the community database at teorth/erdosproblems, which lists the problem as open and unformalized; the formal-conjectures tree; the Zenodo record's version 9 and its companion note; Tao's arXiv preprint through its card.

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.

• bernstein_1931_limitation_values_polynomial_segment
• bernstein_1931_limitation_values_polynomial_segment / equation_34_local_growth
• bernstein_1931_limitation_values_polynomial_segment / source_proof_scope
• bernstein_1931_limitation_values_polynomial_segment / theorem
• erdos_1961_extremal_problem_theory_interpolation
• erdos_1961_extremal_problem_theory_interpolation / theorem_ii
• erdos_1961_problems_results_interpolation_ii
• erdos_1961_problems_results_interpolation_ii / theorem_1
• erdos_1967_problems_results_convergence_divergence_properties_lagrange
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / equations_1_2
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / equations_6_7
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue / corollary_1_11
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue / theorem_1_10_i_transfer