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Source. G. Pisier, Arithmetic characterizations of Sidon sets, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 87--89; notation on pp. 87--88, identity (1) and Theorem 1 on p. 88. The copy read is identified on the source card.

Read depth. Claims checked: the statement, its notation and the remarks after it were read clause by clause on the print. The article is an announcement and contains no proof; nothing here is independently reviewed.

Statement

Let $G$ be a compact abelian group with dual group $\widehat G$. A set $\Lambda\subset\widehat G$ is Sidon when some constant $K$ gives $\sum_\gamma|\widehat f(\gamma)|\le K\|f\|_{C(G)}$ for every trigonometric polynomial $f$ with $\widehat f$ supported by $\Lambda$ (p. 87). For a set $A\subset\widehat G$, $I_A$ is the set of finitely supported families $(\epsilon_\lambda)_{\lambda\in A}$ in $\{-1,0,1\}^A$; for $\gamma\in\widehat G$, $R(\gamma,A)$ is the number of families in $I_A$ with $\gamma=\sum_{\lambda\in A}\epsilon_\lambda\lambda$, and, for each integer $s\ge0$, $R_s(\gamma,A)$ is the number of those with $\sum|\epsilon_\lambda|=s$, so that $R(\gamma,A)=\sum_{s\ge0}R_s(\gamma,A)$ (pp. 87--88).

Theorem 1 (p. 88). Let $\Lambda\subset\widehat G$ with $0\notin\Lambda$. The following are equivalent.

• (i) $\Lambda$ is a Sidon set.
• (ii) There is a number $\theta<1$ such that every finite $A\subset\Lambda$ satisfies

$$\sum_{s\ge0}\frac1{2^s}R_s(0,A)\le2^{\theta|A|}.$$

• (iii) There is a number $\theta<1$ such that every finite $A\subset\Lambda$ satisfies

$$\sup_{\gamma\in\widehat G}R(\gamma,A)\le3^{\theta|A|}.$$

• (iv) There is a number $\theta<1$ such that every finite $A\subset\Lambda$ satisfies

$$\Bigl\{\sum_{\gamma\in\widehat G}R(\gamma,A)^2\Bigr\}^{1/2}\le3^{\theta|A|}.$$

The print places no lower bound on $\theta$ and gives no dependence of $\theta$ on the Sidon constant. Without any restriction the three quantities are at most $2^{|A|}$, $3^{|A|}$ and $3^{|A|}$ respectively, since $\sum_{\gamma}R(\gamma,A)=3^{|A|}$ and at most $\binom{|A|}{s}2^s$ families have $\sum|\epsilon_\lambda|=s$; each condition therefore asks for a uniform exponential saving. That comparison is this page's, not the paper's.

Proof pointer

The article defers the proof to Pisier's "Condition d'entropie et caractérisations arithmétiques des ensembles de Sidon" (its reference [5], then to appear in the proceedings of the 1982 Torino/Milano conference on modern topics in harmonic analysis), and says the proof relies heavily on his "De nouvelles caractérisations des ensembles de Sidon" (reference [4], Advances in Math. Supplementary Studies 7B (1981), 685--726) and on the Proposition of p. 88. It notes two steps (p. 88): (iii) and (iv) are easily equivalent because $\sum_{\gamma}R(\gamma,A)=3^{|A|}$; and (i) implies (ii) through the identity (1),

$$\prod_{\lambda\in A}\bigl[1+\delta(\lambda+\overline\lambda)\bigr] =\sum_{\gamma\in\widehat G}\gamma\Bigl(\sum_{s\ge0}\delta^sR_s(\gamma,A)\Bigr) \qquad(\delta>0,\ A\subset\Lambda\text{ finite}),$$

taken at $\delta=1/2$, together with integrability properties of $\sum_{\lambda\in A}\operatorname{Re}\lambda$.

Dependencies

Pisier's references [4] and [5] above; the Proposition of p. 88. The article says that Drury's result, that being Sidon is determined by the set of $\{-1,0,1\}$ relations, follows as a corollary of its explicit arithmetic characterizations (p. 87).

Bears on

• Problem 774: the conditions give relation-count reformulations of Sidonicity, which Theorem 2 identifies with proportionate dissociation for infinite subsets of the positive integers. They decide nothing about the finite-union question the problem asks.