---

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 1.2 (p. 3). "There is a constant $\tau>0$ such that if $\{a_1<\ldots<a_n\}\subset\{1,\ldots,N\}$ and $n>(1-\tau)N$, then"

$$M(a_1,\dots,a_n)>\exp\tau n\qquad(1.12).$$

The constant $\tau$ is absolute and not made explicit. The paper presents this as a generalization of the remark of Erdős and Szekeres that $\lim_{n\to\infty}[M(1,2,\dots,n)]^{1/n}$ exists and lies between $1$ and $2$ (1.13); the restatement in section 3 (p. 11) says "strictly between". By Proposition 1.1 the conclusion fails for sets of density about $1/2$.

Section 3 states and proves the result as Proposition 3.1 (p. 11) in a slightly different form: there is a constant $\tau>0$ such that if $S\subset\{1,\dots,n\}$ satisfies $|S|>(1-\tau)n$ then $\log M(S)>cn$ for some $c>0$ (3.3). There $n$ is the size of the ambient interval, not of $S$, and the constant in the conclusion is a separate $c$; the two forms agree after shrinking the constants.

Source. J. Bourgain and M.-C. Chang, On a paper of Erdős and Szekeres, J. Anal. Math. 136 (2018), 253--271; Proposition 1.2 on p. 3 and Proposition 3.1 on p. 11 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 1.2 and 3.1 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 3.1 (pp. 11--13) bounds the maximum below by an average: by convexity of the exponential (Fact 2, p. 4), the sup norm of the product is at least the exponential of minus the minimum over $\theta$ of the cosine series $\sum_{a}\sum_k\cos(2\pi ka\theta)/k$, which by Fact 1 equals $-\log\prod|1-e(a\theta)|$, smoothed by a probability measure $\mu$. With $\mu$ the Fejér kernel of order $nR$ for a large constant $R$, the missing elements of $S$ cost at most $\tau(\log k_0)n$ in the frequencies $k\le k_0$ and the tail $k>k_0$ costs at most $Rn/k_0$; at $\theta=3/(4n)$ the full interval $\{1,\dots,n\}$ contributes $cn-\log k_0$ from the frequencies $k\le k_0$. Choosing $k_0$ large and then $\tau$ small leaves a lower bound $cn/2$.

Dependencies

Facts 1 and 2 of the paper (p. 4); otherwise none beyond the Fejér kernel and the Dirichlet kernel identities, which the proof uses directly.

Bears on

• Problem 256: the proposition bounds the product's maximum below only for exponent sets that fill all but a $\tau$ proportion of an interval $\{1,\dots,N\}$. It gives no lower bound for $f(n)$ or $f_*(n)$, which minimize over all exponent sets, and bears on the problem only in that, since $f_*(n)<\exp\{O(n^{1/2}\log n)\}$ (1.7), the sets of distinct exponents attaining $f_*(n)$ for large $n$ cannot be of this kind.