Erdős Problem #1132

The question

erdos_1132.parts.i. For $x_1,\dots ,x_n\in [-1,1]$ let $$l_k(x)=\frac{{\prod }_{i\ne k}(x-x_i)}{{\prod }_{i\ne k}(x_k-x_i)},$$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\ne k$.

Let $x_1,x_2,\dots \in [-1,1]$ be an infinite sequence, and let $$L_n(x)=\sum _{1\le k\le n}|l_k(x)|,$$ where each $l_k(x)$ is defined above with respect to $x_1,\dots ,x_n$.

Must there exist $x\in (-1,1)$ such that $$L_n(x)>\frac{2}{\pi }\log n-O(1)$$ for infinitely many $n$?

This is the first part, asked in [Er67, p. 68] (see also [Va99, 2.43]); [Er61c, (4)] states it as "quite possibly true". Both papers consider an arbitrary triangular array of nodes; as on the problem page, we take the first $n$ terms of one sequence, a special case. The nodes are required to be distinct, so that $l_k$ is defined. We let the constant $C$ in $O(1)$ depend on the sequence and on $x$, the weakest reading. In [Er61c, (4)] it is an absolute constant.

erdos_1132.parts.ii. For $x_1,\dots ,x_n\in [-1,1]$ let $$l_k(x)=\frac{{\prod }_{i\ne k}(x-x_i)}{{\prod }_{i\ne k}(x_k-x_i)},$$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\ne k$.

Let $x_1,x_2,\dots \in [-1,1]$ be an infinite sequence, and let $$L_n(x)=\sum _{1\le k\le n}|l_k(x)|,$$ where each $l_k(x)$ is defined above with respect to $x_1,\dots ,x_n$.

Is it true that $$\underset{n\to \infty }{\mathrm{limsup}}\frac{L_n(x)}{\log n}\ge \frac{2}{\pi }$$ for almost all $x\in (-1,1)$?

This is the second part, asked in [Er67, p. 68] ("Perhaps (7) holds for almost all $x_0$"), again for triangular arrays. The nodes are required to be distinct. We write the $\mathrm{limsup}$ condition as: for every $\epsilon >0$, there are infinitely many $n$ with $$L_n(x)\ge \left(\frac{2}{\pi }-\epsilon \right)\log n.$$ This is equivalent, and avoids the junk value of a real limsup of an unbounded sequence.

The problem page says that a result of Bernstein [Be31] implies that the set of such $x$ is everywhere dense. In [Er67, p. 68] Erdős derives this from his own unpublished bound (6), which sharpens a result of Bernstein. It also follows from erdos_1132.variants.tao with $\omega (n)=\log \log n$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1132» (2 statements).

theorem erdos_1132.parts.i :
    ∀ x : ℕ → ℝ, (∀ k, x k ∈ Set.Icc (-1 : ℝ) 1) → Function.Injective x →
      ∃ t ∈ Set.Ioo (-1 : ℝ) 1, ∃ C : ℝ, ∃ᶠ n : ℕ in atTop,
        2 / Real.pi * Real.log n - C < lebesgueFunction n x t

theorem erdos_1132.parts.ii :
    ∀ x : ℕ → ℝ, (∀ k, x k ∈ Set.Icc (-1 : ℝ) 1) → Function.Injective x →
      ∀ᵐ t ∂(volume.restrict (Set.Ioo (-1 : ℝ) 1)),
        ∀ ε > (0 : ℝ), ∃ᶠ n : ℕ in atTop,
          (2 / Real.pi - ε) * Real.log n ≤ lebesgueFunction n x t

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1132. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• erdosproblems.com/1132
• [Be31] Bernstein, S., *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. 12 (1961), 235-244. PDF
• [Er67] Erdős, P., *Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems*. Mathematica (Cluj) 10 (1968), 65-73. PDF
• [Ta26b] Tao, T., 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).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.