Erdős Problem #1132

The problem

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 $x\in (-1,1)$?

Statement quoted from erdosproblems.com/1132, maintained by Thomas Bloom. Original source: [Er67,p.68][Va99,2.43]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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