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

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

Statement

Proposition (p. 88). Let $G$ be a compact abelian group and $\Lambda\subset\widehat G$ with $0\notin\Lambda$, as in Theorem 1. The conditions (i)--(iv) of Theorem 1 are also equivalent to each of the following.

• (v) There are numbers $\alpha>0$ and $\rho<1$ such that every finite $A\subset\Lambda$ satisfies

$$m\Bigl(\Bigl\{t\in G\ \Bigm|\ \inf_{\lambda\in A}\operatorname{Re}\lambda(t)>\rho\Bigr\}\Bigr) \le2^{-\alpha|A|}.$$

• (vi) There is a number $\alpha>0$ such that, for every finite $A\subset\Lambda$, there are points $t_1,\dots,t_N\in G$ with $N\ge2^{\alpha|A|}$ and $\sup_{\lambda\in A}|\lambda(t_i)-\lambda(t_j)|\ge\alpha$ for all $i\ne j$.

The print writes $m$ without defining it on these pages; this page reads it as the Haar probability measure of $G$.

Proof pointer

The article says the Proposition is proved in Pisier's "Condition d'entropie et caractérisations arithmétiques des ensembles de Sidon" (its reference [5], then to appear), that the equivalence of (v) and (vi) is formal, and that the implication (v) $\Rightarrow$ (i) answers affirmatively Problem 8.3 of his "De nouvelles caractérisations des ensembles de Sidon" (reference [4], Advances in Math. Supplementary Studies 7B (1981), 685--726) (p. 88).

Dependencies

Pisier's references [4] and [5] above.

Bears on

• Problem 774: context only. Through Theorem 2, (v) and (vi) are further equivalent forms of proportionate dissociation for infinite subsets of the positive integers; they decide nothing about the finite-union question.