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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).

Definition (p. 3, display (1.14); restated on p. 13). A set $D=\{\nu_1,\dots,\nu_m\}\subset\mathbb Z$ is dissociated when it admits no nontrivial relation with coefficients $0,1,-1$: if $\varepsilon_1\nu_1+\cdots+\varepsilon_m\nu_m=0$ with every $\varepsilon_i\in\{0,1,-1\}$, then $\varepsilon_1=\cdots=\varepsilon_m=0$. (The print of (1.14) reads "$\varepsilon_1=0,1,-1$" [sic] for the range of each coefficient; the restatement on p. 13 has $\varepsilon_i$.) The paper notes on p. 14 that Hadamard lacunary sets are dissociated.

Proposition 1.3 (p. 3). If $\{a_1<\cdots<a_n\}$ contains a dissociated set of size $m$, then

$$\log M(a_1,\dots,a_n)\gg\frac{m^{\frac12-\varepsilon}}{(\log n)^{1/2}} \qquad(1.15).$$

The print leaves $\varepsilon$ unquantified; the restatement in section 4 replaces $m^{\frac12-\varepsilon}$ by $m^{\frac12-o(1)}$. The paper notes that (1.15) improves the Erdős--Szekeres bound $f(n)\ge\sqrt{2n}$ (1.3) as soon as $m\gg(\log n)^{3+\varepsilon}$ (1.16).

Section 4 states and proves the result as Proposition 4.1 (p. 14): if $S=\{a_1,\dots,a_n\}$ (printed "$\{a,\dots,a_n\}$" [sic]) contains a dissociated set $D$ of size $m$, then $\log M(a_1,\dots,a_n)\gg m^{\frac12-o(1)}/(\log n)^{\frac12}$ (4.1), which improves the general lower bound of Erdős and Szekeres provided $m>(\log n)^{3+\varepsilon}$. The remark after it (p. 14) recalls that, by a result of Pisier, containing a dissociated set of size $m$ is equivalent to containing a Sidon set in the harmonic-analysis sense of size about $m$, its Sidon constant treated as a constant.

Source. J. Bourgain and M.-C. Chang, On a paper of Erdős and Szekeres, J. Anal. Math. 136 (2018), 253--271; Proposition 1.3 and display (1.14) on p. 3, the definition and Proposition 4.1 on pp. 13--14 of the arXiv version arXiv:1509.08411v2, whose labels and pages are used here; the [[analysis/bourgain_2018_paper_erdos_szekeres/_index|source card]] records the edition. The introduction refers to a §5 for the discussion of dissociated sets; this version has four sections, and that discussion is at the start of section 4.

Read depth. Claims checked: the definition and the statements of Propositions 1.3 and 4.1 were read clause by clause on the page images. The proof (pp. 14--19) was not read beyond its opening reduction (below); nothing here is independently reviewed.

Proof pointer

Since $\int_0^1\log|1-e(a\theta)|\,d\theta=0$ for every nonzero integer $a$, the maximum over $\theta$ of $F(\theta)=\sum_{j=1}^n\log|1-e(a_j\theta)|$ is at least half its $L^1$ norm, so (4.1) follows from the lower bound $\|F\|_1\gg m^{\frac12-o(1)}/(\log n)^{1/2}$ (4.3), which the proof establishes from the expansion $F(\theta)=-\sum_k\frac1kf(k\theta)$ with $f(\theta)=\sum_j\cos2\pi a_j\theta$ (4.4). The rest of the proof, pp. 14--19, ends "This proves (4.3) and hence Proposition 4.1".

Dependencies

Pisier's characterization of Sidon sets (Bull. Amer. Math. Soc. 8 (1983), 87--89) is cited in the remark after the proposition; whether the proof uses it was not checked here.

Bears on

• Problem 256: the proposition bounds the product's maximum below for exponent sets that contain a large dissociated subset, and so improves on the Erdős--Szekeres lower bound for those sets. It gives no lower bound for $f(n)$ or $f_*(n)$, which minimize over all exponent sets, and the paper says the general lower bound "remains unimproved" (p. 13).