Problem 1153

../

claims/: The 3 claim pages of Problem 1153, 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

$$\lambda(x)=\sum_k \lvert l_k(x)\rvert.$$

Is it true that, for any fixed $-1\leq a< b\leq 1$,

$$\max_{x\in [a,b]}\lambda(x)> \left(\frac{2}{\pi}-o(1)\right)\log n?$$

Formulation. The statement above is the site's wording. Its denominators require the nodes to be pairwise distinct; this is the intended ordinary-Lagrange setting. One may reorder them as $-1\le x_1<\cdots<x_n\le1$ without changing $\lambda$. The interval $[a,b]$ has positive length and is fixed independently of $n$.

Status. Proved. The catalog labels the problem PROVED (page last edited 1 April 2026, accessed 2026-10-07), crediting Tao [Ta26b]; the frontmatter standing (solved, proved) derives from the accepted claim page Tao 2026. The status-defining source is Tao's arXiv:2603.21453v3 preprint, Theorem 1.10(i), with the elementary transfer below; no journal acceptance, compilation of the complete source proof or independent review of that proof is recorded, and the site's proof-claims page lists one unexamined alternative proof, the pending claim page Yang 2026 (Formalization). The refereed whole-interval bound of Erdős, the case $a=-1$, $b=1$, has the accepted partial claim page Erdős 1961.

Source. erdosproblems.com/1153, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1153, https://www.erdosproblems.com/1153.

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.
• [ErSz78] Erdős, P. and Szabados, J., On the integral of the Lebesgue function of interpolation. Acta Math. Acad. Sci. Hungar. 32 (1--2) (1978), 191--195. The site's reference record resolves the key [ErSz78] to Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients, Austral. Math. Soc. Gaz. (1978), 97-99, which does not concern this problem; the entry above is the paper the commentary describes, whose integral bound gives the maximum bound $\max_{[a,b]}\lambda\gg\log n$ that the commentary states.
• [ErTu61] Erdős, P. and Turán, P., An extremal problem in the theory of interpolation. Acta Math. Acad. Sci. Hungar. (1961), 221-234.
• [Ta26b] T. Tao, Local Bernstein theory, and lower bounds for Lebesgue constants. arXiv:2603.21453v3 (2026).
• [Va99] Various contributors, Some of Paul’s favorite problems. July 1999 booklet, item 2.44, on the right leaf of PDF p. 5 of Kovač's public scan.

Formalization. No inspected formal artifact or successful build is recorded. Boris Alexeev's lean-proofs repository holds Erdos1153.lean (added 2026-08-20), which declares itself a formalization of Tao's solution and proves the $\varepsilon$ form of the question; it is linked on Tao 2026, and this corpus has not built it. The site's proof-claims page, lists Ethan Yang using GPT-5.6 Sol, submitted 31 August 2026, with an alternative proof claim and a Lean repository link. The site gives no guarantee of the claim's correctness and states that nobody associated with it has examined any part. The claim page Yang 2026 pins the repository's only commit (2026-08-31) and records the repository's own description of its theorem target, toolchain and expected axioms; no reproduction of the build or the axiom report and no audit of the statement is recorded. The claim is unreviewed, with no formal coverage.

Current assessment

The source/result pages linked below record the exact modern theorem, historical statements and bounded transfers. The selected source statements and the three elementary transfers have received independent strong review, retained as the [[../library/polynomials/tao_2026_local_bernstein_theory_lower_bounds_lebesgue/evidence/verify/transfer_review|transfer review]] filed with the Tao source. The copy of the text the review read is not retained in the repository; the current text agrees with the report's description of it, which is a match of description, not of bytes. Proof coverage: the complete source proofs are not compiled in this corpus. The catalog's PROVED label is the recorded acceptance; refereed publication, independent review of the complete source proof and formal verification are not recorded. The claim pages carry the standing: the accepted Tao 2026 page lists the curator's acceptance as its only evidence, the accepted partial Erdős 1961 page records the refereed whole-interval case, and the pending Yang 2026 page records the alternative proof with its Lean development.

The recorded 6 September 2026 API response identifies Tao v3; no broader current-status search is recorded on this page.

The transfer review read this page's frontmatter status and Status paragraph as they stood on 2026-09-15. Since then the Status paragraph has been reworded to name the catalog's PROVED label, Tao's Theorem 1.10(i) as the status-defining source and the three claim pages, including the accepted partial Erdős 1961 page, the Formalization field to describe the pinned repository, and this assessment to record the review's unretained copy, the recorded acceptance and the claim pages; the frontmatter has moved to the claims schema (status: solved, claim: proved in place of status: proved) without a change of meaning. The standing, the Tao source, the statement and the transfer below are unchanged since the review.

Progress

Tao v3 Theorem 1.10(i), equation (1.26), printed/physical p. 9, gives, for each fixed positive-length $I\subseteq[-1,1]$, constants $C_I\ge0,n_I$ such that for all $n\ge n_I$ and all distinct node configurations,

$$\sup_{x\in I}\lambda(x)\ge\frac2\pi\log n-C_I.$$

The constant and threshold may depend on $I$, and are uniform in the nodes. For $I=[a,b]$, each $l_k$ is a polynomial, so $\lambda$ is continuous and its supremum is its maximum on this compact interval. For $n\ge\max\{n_I,2\}$, $\varepsilon_n=(C_I+1)/\log n\to0$ gives

$$\max_{x\in[a,b]}\lambda(x) \ge\frac2\pi\log n-C_I >\left(\frac2\pi-\varepsilon_n\right)\log n.$$

The [[../library/polynomials/tao_2026_local_bernstein_theory_lower_bounds_lebesgue/theorem_1_10_i_transfer|exact source-statement and transfer record]] supplies the complete bounded deduction. Tao’s own proof is not compiled in this corpus. The v3 PDF read for the source card has the arXiv date 22 April 2026; its separate title-page author date is 23 April. The arXiv API response read identifies v3 and does not establish journal acceptance.

The Erdős--Turán source has two layers. Theorem I and (3.12), p. 224, use derivative-data Hermite polynomials $\mathfrak{h}_{jn}=(x-x_{jn})l_{jn}^2$ and the scale $\log n/n$. On p. 225, equation (3.13), called Theorem II, gives the ordinary-Lagrange global bound $(2/\pi)\log n-c_5\log\log n$; the authors then omit analogous ordinary local formulations. That page supplies the historical bridge, while Va99 item 2.44 prints the exact ordinary local question, with $+o(1)$ in place of $-o(1)$. The sign of an unrestricted $o(1)$ term is immaterial.

The separate Erdős 1961 II source proves the ordinary global bound $(2/\pi)\log n-c_1$. Its p. 236 local display is over $a<x<b$, has coefficient $1/4-\varepsilon$, and requires $n>n_0(\varepsilon,a,b)$; the $2/\pi$ replacement is left unproved there. Bernstein 1931 uses degree $n$ and $n+1$ nodes: its full-segment minimax asymptotic and its conditional $1/2$ versus all-cases $1/4$ local estimates are historical context. Its full-segment asymptotic, the case $a=-1$, $b=1$, has no claim page: Erdős (Er61c, p. 235) writes that Bernstein proved the bound in full only for trigonometric interpolation and that he could not reconstruct the algebraic case, and Tao (v3, p. 7) says that Bernstein asserted it without full proof details; the instance is settled by Erdős 1961. Erdős--Szabados 1978, Theorem (4) on p. 191, gives $\int_a^b\lambda(x)\,dx\ge c_3(b-a)\log n$ for $n\ge n_2(a,b)$ and an absolute $c_3>0$. This implies qualitative local growth, without specifying the sharp coefficient.

The Tao source home records the author’s AI disclosures on pp. 12 and 14 and the dated discussion chronology. The proposed February proof gap concerns a manuscript before the 24 March rewrite; it is not assigned to v3 without a version comparison. Source-declared AI roles alone confer no correctness or formal credit.

Problem 1129 is contextual global minimax interpolation. Problem 1132 is the distinct pointwise question; Tao Corollary 1.11 gives a dense set of points and an infinite subsequence with divergent loss $\omega(n)$, not its stronger target. Neither related problem receives a status transfer here.

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.

• various_1999_some_pauls_favorite_problems
• various_1999_some_pauls_favorite_problems / problem_2_44
• bernstein_1931_limitation_values_polynomial_segment
• bernstein_1931_limitation_values_polynomial_segment / equation_27_midpoint_product
• bernstein_1931_limitation_values_polynomial_segment / equation_32_telescoping
• bernstein_1931_limitation_values_polynomial_segment / equation_33_interior_case
• bernstein_1931_limitation_values_polynomial_segment / equation_34_local_growth
• bernstein_1931_limitation_values_polynomial_segment / equations_29_31_logarithmic_pairs
• bernstein_1931_limitation_values_polynomial_segment / interpolation_extremal_identity
• bernstein_1931_limitation_values_polynomial_segment / local_gap_test_companion
• bernstein_1931_limitation_values_polynomial_segment / perturbed_chebyshev_nodes
• 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_extremal_problem_theory_interpolation / two_interpolation_layers
• 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_6_7
• erdos_szabados_1978_integral_lebesgue_function_interpolation
• erdos_szabados_1978_integral_lebesgue_function_interpolation / integral_lower_bound
• erdos_szabados_1978_integral_lebesgue_function_interpolation / node_gap_lemma
• erdos_turan_1940_on_interpolation_iii
• erdos_turan_1940_on_interpolation_iii / lemma_iv_adjacent_fundamental_polynomials
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue / theorem_1_10_i_transfer
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue / theorem_1_10_ii
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue / theorem_1_13
• tao_2026_local_bernstein_theory_lower_bounds_lebesgue / theorem_1_6