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Statement

Theorem XI (p. 544). Let $p(x)\sqrt{1-x^2}$ be continuous in $[-1,1]$ with $p(x)\sqrt{1-x^2}\ge m>0$, and let $C(n)$ tend to infinity, arbitrarily slowly, as $n\to\infty$. Then for the roots $\cos\vartheta_\nu^{(n)}$ of the $n$th polynomial orthogonal to $p(x)$ that satisfy

$$\frac{C(n)}{n}\le\vartheta_k^{(n)}<\vartheta_{k+1}^{(n)}\le\pi-\frac{C(n)}{n},$$

one has, uniformly in $k$, $\lim_{n\to\infty}n(\vartheta_{k+1}^{(n)}-\vartheta_k^{(n)})=\pi$.

Remark II (p. 545) adds that the theorem in this form does not hold for every root; the paper says the gap is then asymptotically the distance from $\vartheta_k^{(n)}$ to the nearest root on its right of $\cos(n-1)\vartheta_k^{(n)}\cos n\vartheta-\cos n\vartheta_k^{(n)}\cos(n-1)\vartheta=0$, without proof.

Proof pointer

P. 544: the paper deduces it "easily" from Theorem IX and Lemma VIII (p. 539): if $n\varphi_0\to\infty$ and $n(\pi-\varphi_0)\to\infty$, the root of $\phi_{n-1}(\cos\vartheta)=0$ nearest to $\vartheta=\varphi_0$ is at distance $\sim\pi/n$. No further details are printed.

Read depth

Claims checked: Theorem XI, Lemma VIII and Remark II were read clause by clause on the page images of the print. The deduction is not written out in the paper and was not reconstructed here. Nothing here is independently reviewed.

Dependencies

Theorem IX and Lemma VIII of the same paper.

Source. P. Erdős and P. Turán, On interpolation. III. Interpolatory theory of polynomials, Annals of Mathematics (2) 41 (3) (1940), 510--553, DOI 10.2307/1968733; the edition read is named on the source card.

Bears on

None of the problem pages directly.