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Updated Oct 8

Problem 256

claims/: The 1 claim page of Problem 256, one per claimant's result; the problem's standing derives from them.

Statement. Let $n\geq 1$ and $f(n)$ be maximal such that for any integers $1\leq a_1\leq \cdots \leq a_n$ we have

\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).

Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that

\log f(n) \gg n^c?

Status. Open, the site's label (OPEN; page last edited 20 January 2026). The site's commentary answers the specific question no: Belov and Konyagin proved $\log f(n)\ll(\log n)^4$ (their 1996 paper, an accepted partial claim on its refereed publication). Estimating $f(n)$ remains open.

Source. erdosproblems.com/256, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #256, https://www.erdosproblems.com/256.

References.

[At61] Atkinson, F. V., On a problem of Erdős and Szekeres. Canad. Math. Bull. (1961), 7-12.
[BeKo96] Belov, A. S. and Konyagin, S. V., An estimate for the free term of a nonnegative trigonometric polynomial with integer coefficients. Mat. Zametki (1996), 627-629. The site's key names this short note, Mat. Zametki 59 (1996), no. 4, 627-629 (Math. Notes 59 (1996), no. 4, 451-453), which bounds the least constant term of a nonnegative cosine polynomial with integer coefficients (card). The bound $\log f(n)\ll(\log n)^4$ that the site's commentary credits to [BeKo96] is in the authors' paper 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). Tang's Proc. Amer. Math. Soc. paper cites that paper for the bound, and the claim page records it.
[BoCh18] Bourgain, J. and Chang, Mei-Chu, On a paper of Erdős and Szekeres. J. Anal. Math. (2018), 253-271.
[ErSz59] Erdős, P. and Szekeres, G., On the product $\prod_{k=1}^n(1-z^{a_k})$ . Acad. Serbe Sci. Publ. Inst. Math. (1959), 29-34.
[Od82] Odlyzko, A. M., Minima of cosine sums and maxima of polynomials on the unit circle. J. London Math. Soc. (2) (1982), 412-420.
[Ta26] Tang, Quanyu, An improved lower bound for Erdős–Szekeres products. Proc. Amer. Math. Soc. 154 (2026), no. 8, 3381-3388, DOI 10.1090/proc/17668; arXiv:2509.14182. Not a site reference; added for the lower bound $f(n)\ge2\sqrt n$ .

Formalization. None recorded.

Current assessment

The site formulation asks for two things: an estimate of $f(n)$ , the least over all choices of $n$ exponents $1\le a_1\le\cdots\le a_n$ of the maximum modulus on the unit circle of $\prod_i(1-z^{a_i})$ , and whether $\log f(n)\gg n^c$ for some $c>0$ . The site labels the problem OPEN (page last edited 20 January 2026), and its commentary answers the second question no. The standing targets the question as stated.

Upper bounds. Erdős and Szekeres [ErSz59] (card) proved $\lim f(n)^{1/n}=1$ , so $f(n)$ grows more slowly than any exponential, and noted the bound $f(n)>\sqrt{2n}$ . Erdős proved by probabilistic methods that $\log f(n)\ll n^{1-c}$ for some $c>0$ , as the site's commentary records. Atkinson [At61] (card) proved $\log f(n)\ll n^{1/2}\log n$ , and Odlyzko [Od82] improved this to $\log f(n)\ll n^{1/3}(\log n)^{4/3}$ . Belov and Konyagin proved $\log f(n)\ll(\log n)^4$ , in the Izvestiya paper that the References annotate; this answers the second question no, and is recorded as an accepted partial claim on their claim page, on its refereed publication alone, since the site's label is OPEN. For the variant $f^*(n)$ with distinct exponents $a_1<\cdots<a_n$ , Bourgain and Chang [BoCh18] (card) proved in Proposition 1.1 that some set of $n$ distinct exponents in $\{1,\ldots,N\}$ , with $n$ comparable to $N/2$ , has product maximum at most $\exp(c(n\log n)^{1/2}\log\log n)$ . This gives $\log f^*(n)\ll(n\log n)^{1/2}\log\log n$ only at the sizes $n$ the construction produces; the paper states no bound for $f^*(n)$ at every $n$ . [BoCh18] also recalls, as Atkinson's relation (1.9), a link between $f^*$ and the Chowla cosine problem, Problem 510: if every set $A$ of $n$ integers has a $\theta$ with $\sum_{a\in A}\cos(a\theta)<-M_n$ , then $\log f^*(n)\ll M_n\log n$ . Atkinson's 1961 paper [At61] states no such result: its closing remarks (pp. 11--12) only suggest, for $f(n)$ with repeated exponents allowed, a connection with the minimum of a sum of cosines.

Lower bounds. The bound $f(n)\ge\sqrt{2n}$ of Erdős and Szekeres stood until Tang [Ta26] proved $f(n)\ge2\sqrt n$ in a refereed paper. The improvement settles no instance of the problem, since it fixes neither the order of $f(n)$ nor an answer to either question, so it has no claim page.

Search scope, 2026-10-07: the site's page and commentary (last edited 20 January 2026), the community database (the problem recorded as unformalized), the formal-conjectures catalog (no statement file for the problem) and the arXiv and Crossref records of Tang's paper. No proof claim is recorded for the problem. Remaining gaps: the order of $f(n)$ , between $2\sqrt n$ and $\exp(C(\log n)^4)$ ; no proof has been compiled or independently reviewed by this corpus.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (22)

Belov and Konyagin's polylogarithmic upper bound problems/analysis/E0256/claims Analysis analysis/atkinson_1961_problem_erdos_szekeres Inequality (5): log f(n) <= n^(1/2)((1/2) log n + 4 log 2)Lemma 1: a one-sided bound for log|1 - e^(i theta)| by a weighted partial Fourier sum Lemma 2: N(c_1,...,c_p) <= c_0 log M + 2M^(-1) log 2 sum c_k under a non-negative cosine polynomial analysis/belov_1996_estimate_free_term_nonnegative_trigonometric_polynomial Corollary 1: (ln n)^2/ln ln n << K(n) << M(n) << (ln n)^3 for n >= 3 Corollary 2: ln f(n) << (ln n)^4 for the Erdős–Szekeres products, n >= 2 Theorem 1: (n^q) is strictly admissible for 1 < q <= 2; lacunary sequences are not admissible Theorem 2: the free-term minima M and K compared with the admissible-sequence functional Phi Theorem 3: two random sequences are strictly admissible with probability above 1/2 Theorem 4: (ln x)^2/ln ln x << Phi(x) << (ln x)^3 for x >= 3 analysis/bourgain_2018_paper_erdos_szekeres Proposition 1.1 (p. 2): about N/2 distinct exponents in [1, N] with product maximum at most exp(c root n root log n log log n)Proposition 1.2 (p. 3): an almost full exponent set in [1, N] has product maximum exponentially large Proposition 1.3 (p. 3): a dissociated subset of size m forces log M at least of order m to the 1/2 - epsilon over root log n analysis/erdos_1959_product Theorem 1 (pp. 31-32): the product of the distances from 1 of the points t alpha on the circle is subexponential unless alpha is near a small-denominator rational Theorem 2 (p. 33): the n-th root of f(n) tends to 1 Theorem 3 (p. 34): f(n) is at least the square root of 2n

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