---

Statement

For positive integers $a_1<\cdots<a_n$ let $M(a_1,\dots,a_n)=\max_{|z|=1}\prod_{i=1}^n|1-z^{a_i}|$ (display (1.1), p. 1).

Proposition 1.1 (p. 2). There is a subset $\{a_1<\cdots<a_n\}\subset\{1,\dots,N\}$ with $n\asymp N/2$ such that

$$M(a_1,\dots,a_n)<\exp\bigl(c\sqrt n\sqrt{\log n}\,\log\log n\bigr) \qquad(1.11).$$

The print leaves the quantifier on $N$ implicit (the statement is read as holding for each large $N$) and does not make the constant $c$ explicit. The paper says the proposition improves upon Kolountzakis's construction (1.7), which has $1<a_1<\cdots<a_n<2n+O(\sqrt n)$ and $M(a_1,\dots,a_n)<\exp\{O(n^{1/2}\log n)\}$.

The same result is stated and proved in section 2 as Proposition 2.2 (p. 5), with the roles of the letters exchanged: a subset $\{a_1,\dots,a_m\}\subset\{1,\dots,n\}$ of size $m\asymp n/2$ with $\bigl|\prod_{k=1}^m|1-z^{a_k}|\bigr|_{L^\infty(|z|=1)}\le e^{c\sqrt n\sqrt{\log n}(\log\log n)}$ (2.4). The remark after it (p. 5) calls (2.4) a slight improvement of the bound $e^{c\sqrt n\log n}$ that follows from a construction of Kolountzakis (Proc. Amer. Math. Soc. 120 (1994), p. 162) together with Lemma 2.1.

Source. J. Bourgain and M.-C. Chang, On a paper of Erdős and Szekeres, J. Anal. Math. 136 (2018), 253--271; Proposition 1.1 on p. 2 and Proposition 2.2 on p. 5 of the arXiv version arXiv:1509.08411v2, whose labels and pages are used here; the source card records the edition.

Read depth. Claims checked: the statements of Propositions 1.1 and 2.2 and the remark after 2.2 were read clause by clause on the page images. The proof was read for its structure only (below); no step was checked, and nothing here is independently reviewed.

Proof pointer

The proof of Proposition 2.2 (pp. 6--10) chooses the exponents at random: independent $0,1$ selectors $\xi_j$, $1\le j<n$, with mean $1-j/n$, so that the expected cosine sum of the chosen set is a Fejér kernel (2.5). Lemma 2.1 (p. 4) bounds the logarithm of the product above by a weighted cosine sum; the mean part contributes at most $\log J$ because the Fejér kernel is nonnegative, and the random part is bounded with large probability by the probabilistic Salem--Zygmund inequality (2.10), which leaves a square sum (2.12) to estimate. That sum is bounded by $O(n(\log\log n)^2)$ for every $\theta$ except those with $|\ell\theta-\ell a/q|<e^{-\sqrt q}$ for $1\le\ell\le n$ and a small denominator $q$ (2.20), which are handled by comparing the random product with its expectation and evaluating the resulting product over residues mod $q$ with Lemma 2.1 again (displays (2.21)--(2.26), pp. 9--10).

Dependencies

Lemma 2.1 (p. 4), whose proof rests on a calculation in Odlyzko's Proposition 1 (J. London Math. Soc. (2) 26 (1982), 412--420), display (2.4) there; the probabilistic Salem--Zygmund inequality, cited from Kolountzakis's survey (Number theory, New York Seminar 1991--1995, Springer, 1996, 229--251). Neither was checked here.

Bears on

• Problem 256: the proposition gives sets of distinct exponents whose maximum is at most $\exp(c\sqrt n\sqrt{\log n}\log\log n)$, and since $f(n)\le f_*(n)\le M(a_1,\dots,a_n)$ it bounds $f_*(n)$, and hence $f(n)$, from above for the sizes $n$ it produces. The paper states no new bound for $f_*(n)$ at every $n$. It does not touch the question whether $\log f(n)\gg n^c$, which the bound $\log f(n)\ll(\log n)^4$ of Belov and Konyagin answers.