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Statement

Theorem XV (p. 552). Suppose that for the matrix $\mathfrak M$

$$|l_k(x)|\le c_{87}n^{c_{88}},\qquad-1\le x\le1,\quad k=1,\ldots,n,\quad n=1,2,\ldots.$$

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

$$\Bigl|\sum_{\alpha\le\vartheta_\nu^{(n)}\le\beta}1-\frac{\beta-\alpha}{\pi}n\Bigr| <c_{89}(c_{87},c_{88},\epsilon)\,n^{1/2+\epsilon}.$$

The introduction (p. 519) announces this as (24)--(25), uniform distribution already for intervals of length $1/n^{\frac12-2\epsilon}$.

Proof pointer

Pp. 552--553. The paper says it suffices to prove the upper estimate for every subinterval, since applying it to $[0,\alpha]$ and $[\beta,\pi]$ gives the lower estimate; the upper estimate is proved "completely analogous" to that of Theorem XIV.

Read depth

Claims checked: Theorem XV and the reduction on pp. 552--553 were read clause by clause on the page images of the print. The upper estimate is not written out in the paper and was not reconstructed here. Nothing here is independently reviewed.

Dependencies

The method of Theorem XIV.

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.