Erdős Problem #1130

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 $x_0=-1$ and $x_{n+1}=1$ and

$$\Upsilon(x_1,\ldots,x_n)=\min_{0\leq i\leq n}\max_{x\in[x_i,x_{i+1}]} \sum_k \lvert l_k(x)\rvert.$$

Is it true that

$$\Upsilon(x_1,\ldots,x_n)\ll \log n?$$

Describe which choice of $x_i$ maximise $\Upsilon(x_1,\ldots,x_n)$.

Statement quoted from erdosproblems.com/1130, maintained by Thomas Bloom. Original source: [Er47,p.1172][Er67,p.66]. Current status there: proved.

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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