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on Erdős #1132 · analysis / polynomials

The 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 $x_1,x_2,\ldots\in [-1,1]$ 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 $x\in (-1,1)$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.