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Claim. There is an absolute constant $C$ such that for every $n\ge2$ some integers $1\le a_1\le\cdots\le a_n$ satisfy

$$\max_{|z|=1}\Bigl|\prod_{i=1}^n(1-z^{a_i})\Bigr|\le\exp\bigl(C(\log n)^4\bigr).$$

Hence the quantity $f(n)$ of Problem 256 satisfies $\log f(n)\ll(\log n)^4$, and $\log f(n)\gg n^c$ fails for every $c>0$. This is the bound of A. S. Belov and S. V. Konyagin, An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients, Izv. Math. 60 (1996), no. 6, 1123--1182 (Russian original Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), no. 6, 31--90). It is recorded as the site's commentary states it and as Q. Tang, An improved lower bound for Erdős–Szekeres products, Proc. Amer. Math. Soc. 154 (2026), no. 8, 3381--3388 (arXiv:2509.14182), states it in citing the paper as his reference [3].

Covers. The second question of Problem 256, whether $\log f(n)\gg n^c$ for some $c>0$, answered no. The first question, to estimate $f(n)$, is not settled: the bounds in force are $2\sqrt n\le f(n)\le\exp(C(\log n)^4)$.

Depends on. Nothing in this wiki: the bound is the paper's own.

Acceptance. Refereed: a journal publication, Izvestiya: Mathematics 60 (1996), no. 6, the DOIs linked above (the translation's record dates the issue to 31 December 1996, and the page is named by the issue month, December 1996, filled to the first of the month; the Russian original's record gives only the year and issue). Reviewed is not listed: the site credits the bound in its commentary on a problem it labels OPEN, which is commentary and not acceptance of a solution. Formalized is not listed: no Lean statement or proof of the bound is recorded.