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Erdős #1132 · analysis / polynomials

For x1,…,xn∈[−1,1]x_1,\ldots,x_n\in [-1,1] let

lk(x)=∏i≠k(x−xi)∏i≠k(xk−xi),l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},

which are such that lk(xk)=1l_k(x_k)=1 and lk(xi)=0l_k(x_i)=0 for i≠ki\neq k.

Let x1,x2,…∈[−1,1]x_1,x_2,\ldots\in [-1,1] be an infinite s…

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

For x1,…,xn∈[−1,1]x_1,\ldots,x_n\in [-1,1] let

lk(x)=∏i≠k(x−xi)∏i≠k(xk−xi),l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},

which are such that lk(xk)=1l_k(x_k)=1 and lk(xi)=0l_k(x_i)=0 for i≠ki\neq k.

Let x1,x2,…∈[−1,1]x_1,x_2,\ldots\in [-1,1] be an infinite sequence, and let

Ln(x)=∑1≤k≤n∣lk(x)∣,L_n(x) = \sum_{1\leq k\leq n}\lvert l_k(x)\rvert,

where each lk(x)l_k(x) is defined above with respect to x1,…,xnx_1,\ldots,x_n.

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

Ln(x)>2πlog⁡n−O(1)L_n(x) >\frac{2}{\pi}\log n-O(1)

for infinitely many nn?

Is it true that

lim sup⁡n→∞Ln(x)log⁡n≥2π\limsup_{n\to \infty}\frac{L_n(x)}{\log n}\geq \frac{2}{\pi}

for almost all x∈(−1,1)x\in (-1,1)?

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Background

A result of Bernstein [Be31] implies that the set of x∈(−1,1)x\in(-1,1) for which

lim sup⁡n→∞Ln(x)log⁡n≥2π\limsup_{n\to \infty}\frac{L_n(x)}{\log n}\geq \frac{2}{\pi}

is everywhere dense.

Erdős [Er61c] proved that, for any fixed x1,…,xn∈[−1,1]x_1,\ldots,x_n\in [-1,1],

max⁡x∈[−1,1]∑1≤k≤n∣lk(x)∣>2πlog⁡n−O(1).\max_{x\in [-1,1]}\sum_{1\leq k\leq n}\lvert l_k(x)\rvert>\frac{2}{\pi}\log n-O(1).

Tao [Ta26b] has proved that for any function ω(n)\omega(n) which tends to infinity with nn there exists a dense set of x∈(−1,1)x\in (-1,1) such that

Ln(x)≥2πlog⁡n−ω(n)L_n(x)\geq \frac{2}{\pi}\log n-\omega(n)

for infinitely many nn. Tao also notes that it is unclear in the question of Erdős whether the constant in the O(1)O(1) term may depend on xx.

See also [1129] for more on Ln(x)L_n(x), and also [1153].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1132 link

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