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Source. P. Erdős, Problems and results on the theory of interpolation. II, Acta Math. Acad. Sci. Hungar. 12 (1961), 235--244 (source card): the intervals $I_t$ defined on p. 237 and Theorem 2 on p. 242.

Read depth. Claims checked: the statement and the definition of $I_t$ were read clause by clause on the page images. The paper gives no proof.

Statement

Setting (p. 237). Let $x_0=\cos\vartheta_0$ be the point of $(-1,+1)$ at which $|\omega_n(x)|$ attains its maximum there. For $t>0$, $\bar I_t$ is the intersection with $(0,\pi)$ of an interval of length $t\pi/n$ having $\vartheta_0$ as an endpoint, and $I_t$ is the interval of $(-1,+1)$ obtained from $\bar I_t$ by the map $\cos\vartheta=x$. There are two such intervals, one on each side of $x_0$.

Theorem 2 (p. 242). Let $\omega_n(x)=\prod_{i=1}^n(x-x_i)$, where the $x_i$ need not lie in $(-1,+1)$, and suppose $|\omega_n|$ attains its maximum over $(-1,+1)$ at $x_0=\cos\vartheta_0$. Then every interval $I_t$ contains at most $c_{14}t$ of the $x_i$, where $c_{14}$ is an absolute constant.

The paper does not give the proof. It says the best value of $c_{14}$ is not known and suggests that perhaps $c_{14}=2$ (p. 242). The same symbol $c_{14}$ also names an unrelated constant in the proof of Theorem 1 (p. 241).

Context

The paper states the theorem as a way the proof of Theorem 1 could have been organized differently: that proof treats separately the case in which some $I_{t'}$ contains more than $t'^3$ nodes (Lemma 3, p. 237), and Theorem 2 shows no $I_t$ ever contains that many once $t$ is large.

Bears on

No Erdős problem page states a question this theorem answers.