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Source. Theorem 6, p. 1175, of P. Erdős, "Some remarks on polynomials," Bull. Amer. Math. Soc. 53 (1947), 1169-1176. Pages are the journal's own, as on the source card.

Statement

The quantity $\omega_n(M,z_0)$ and the two open questions it answers are as on the Theorem 5 page.

Theorem 6 (p. 1175). Let $n_1<n_2<\cdots$ tend to infinity sufficiently fast, and let $M$ consist of the point $0$ and the points $1/2^u$ with $n_i\le u\le2n_i+1$ for some $i$. Then $\lim\omega_n(M,0)^{1/n}$ does not exist; in fact

$$\limsup\omega_n(M,0)^{1/n}=\infty,\qquad \liminf\omega_n(M,0)^{1/n}<\infty .$$

The print writes the second limit with $z_0$ in place of $0$. This answers the first open question in the negative.

Read depth. Claims checked: the statement was read clause by clause on the print. The short proof was read; its lim inf half follows Theorem 5, and its lim sup half is garbled as printed (see the proof pointer).

Proof pointer

Page 1175. As in Theorem 5, a polynomial of degree $n_i$ bounded by $1$ at the points $1/2^u$, $n_i\le u\le2n_i+1$, has derivative at $0$ below $c^{n_i}$, which bounds the lim inf. For the lim sup, the print takes a constant multiple of $\prod_k(x-1/2^k)$, over $k=1,\ldots,2n_i+1$, says that its degree is $2n_i+2$ and that it is below $1$ in absolute value on $M$, and states that, when $n_{i+1}$ grows fast enough, the root of order $2n_i+2$ of its derivative at $0$ tends to infinity. The construction is garbled as printed: the product has $2n_i+1$ factors, not $2n_i+2$, and the scan does not show clearly whether the constant factor and the lower bound use $2^{n_i+1}$ or $2^{n_{i+1}}$; read with $2^{n_i+1}$, the printed lower bound tends to $0$. This half of the proof was not checked.