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Statement

Theorem XIII (p. 547). Let $p(x)$ be non-negative and $L$-integrable in $[-1,1]$, and suppose that its roots (zeros) form a set of measure $0$. Then for the roots $\cos\vartheta_\nu^{(n)}$ of the $n$th orthogonal polynomial belonging to $p$,

$$\lim_{n\to\infty}\frac1n\sum_{\alpha\le\vartheta_\nu^{(n)}\le\beta}1 =\frac{\beta-\alpha}{\pi}$$

for every fixed subinterval $[\alpha,\beta]$ of $[0,\pi]$.

The introduction (pp. 514, 519--520) compares this with Szegő's sufficient condition, integrability of $\log p(x)/\sqrt{1-x^2}$, and states that the first author has shown, with the proof omitted, that the necessary and sufficient condition for uniform distribution involves the transfinite diameter of the zero set of $p$.

Proof pointer

P. 547: Lemma VII (stated on the Theorem VI page) supplies the hypothesis of Theorem XII.

Read depth

Claims checked: Theorem XIII and the comparison in the introduction were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Theorem XII and Lemma VII 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.