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Statement

Question (4) (p. 30). The paper asks whether, for every $\alpha$,

$$\limsup_{n\to\infty}\ \max_{\lvert z\rvert=1}\ \prod_{k=1}^n \bigl\lvert z-e^{2\pi ik\alpha}\bigr\rvert=\infty.\qquad(4)$$

Conjecture (p. 30, unnumbered). The paper records, as an old conjecture of P. Erdős which would imply (4), the following: if $z_1,z_2,\ldots$ is any infinite sequence with $\lvert z_i\rvert=1$, then

$$\limsup_{n\to\infty}\ \max_{\lvert z\rvert=1}\ \prod_{i=1}^n \lvert z-z_i\rvert=\infty.$$

Accompanying remarks (p. 30). For the denominators $q_n$ of the rational approximations $\lvert\alpha-p_n/q_n\rvert=o\bigl(1/(q_n^2\log q_n)\bigr)$, which Khintchine's theorem supplies for almost all $\alpha$ (display (2)), the paper says a simple computation gives

$$\lim_{n\to\infty}\ \max_{\lvert z\rvert=1}\ \prod_{k=1}^{q_n} \bigl\lvert z-e^{2\pi ik\alpha}\bigr\rvert=2.\qquad(5)$$

It suggests, without proof, that perhaps the corresponding liminf over all $n$ is finite for every irrational $\alpha$, and notes that it is infinite for rational $\alpha$. The paper proves none of these statements.

Source. P. Erdős and G. Szekeres, On the product $\prod_{k=1}^n(1-z^{a_k})$, Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), 29--34: question (4), the conjecture and display (5) on p. 30. The edition read is identified on the source card.

Read depth. Claims checked: the question, the conjecture and the remarks were read clause by clause on the printed page. The paper gives no proof, so none was checked. Nothing here is independently reviewed.

Proof pointer

None in the paper; the statements are posed as questions.

Dependencies

None.

Bears on

• Problem 119: the conjecture recorded here is the problem's first question, whether $\limsup M_n=\infty$ for every sequence $z_i$ on the unit circle. The paper states it as an old conjecture of Erdős and proves nothing about it.