Problem 1132

Erdős Problem #1132: declared status 'open'. Formalized: no. 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 $x\in (-1,1)$? Current best: Erd\H{o}s [Er61c] proved that, for any fixed $x_1,\ldots,x_n\in [-1,1]$,$$\max_{x\in [-1,1]}\sum_{1\leq k\leq n}\lvert l_k(x)\rvert>\frac{2}{\pi}\log n-O(1).$$See also [1129] for more on $L_n(x)$. Prize: no. Tags: analysis, polynomials.