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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 Definition of Rider and quasi-independent sets on p. 88, the Problem and Theorem 2 on p. 89. The copy read is identified on the source card.

Read depth. Claims checked: the definitions, the statement and the remarks around it were read clause by clause on the print. The article proves only the direction from (vii) to Sidon, by citation; nothing here is independently reviewed.

Statement

Let $G$ be a compact abelian group with dual group $\widehat G$, let $R(0,\Lambda)$ count the finitely supported families $(\epsilon_\lambda)_{\lambda\in\Lambda}$ in $\{-1,0,1\}^\Lambda$ with $\sum_{\lambda\in\Lambda}\epsilon_\lambda\lambda=0$, and let $R_s(0,\Lambda)$ count those among them with $\sum|\epsilon_\lambda|=s$.

Definition (p. 88). $\Lambda\subset\widehat G$ is quasi-independent if $R(0,\Lambda)=1$, equivalently if $R_s(0,\Lambda)=0$ for all $s\ge1$: the only relation $\sum_{\lambda}\epsilon_\lambda\lambda=0$ with coefficients in $\{-1,0,1\}$ is the trivial one. $\Lambda$ is a Rider set if $\sum_{s\ge0}\delta^sR_s(0,\Lambda)<\infty$ for some $\delta>0$.

Theorem 2 (p. 89). "A subset $\Lambda$ of $\widehat G$ is a Sidon set iff (vii) there is an integer $k$ such that any finite subset $A$ of $\Lambda$ contains a quasi-independent subset $B\subset A$ with $|B|\ge|A|/k$."

Theorem 2 as printed does not repeat Theorem 1's hypothesis $0\notin\Lambda$. This page notes, as its own remark rather than the paper's, that a set containing $0$ fails (vii) at $A=\{0\}$, since $\{0\}$ is not quasi-independent, although finite sets are Sidon; so the statement is to be read with $0\notin\Lambda$, which holds in the integer case below. The abstract (p. 87) states the same result with a number $\delta>0$ and $|B|\ge\delta|A|$ in place of $1/k$.

Integer case. This paragraph is the corpus's translation, not the paper's. For $\widehat G=\mathbb Z$ (so $G=\mathbb T$) and $\Lambda\subset\mathbb N$, quasi-independence is dissociation in the sense of Problem 774: a nonzero relation splits its support into the coefficient-$1$ and coefficient-$(-1)$ parts, two distinct finite subsets with equal sums, and conversely two distinct finite subsets with equal sums give, after removing their intersection, a nonzero relation. A size bound $\ge c|B|$ with a constant $c>0$ gives (vii) with any integer $k\ge1/c$, and (vii) gives it with $c=1/k$. So an infinite $\Lambda\subset\mathbb N$ is proportionately dissociated in that problem's sense if and only if it is a Sidon set.

Proof pointer

The article says the proof that Sidon sets satisfy (vii) is given in Pisier's "Condition d'entropie et caractérisations arithmétiques des ensembles de Sidon" (its reference [5], then to appear), and that the converse follows from Theorem 2.3 of his "De nouvelles caractérisations des ensembles de Sidon" (reference [4], Advances in Math. Supplementary Studies 7B (1981), 685--726), because every quasi-independent set $B$ is Sidon with Sidon constant $S(B)$ bounded by an absolute constant (p. 89).

The article also records (p. 89) that a union of $k$ quasi-independent sets satisfies (vii) with that $k$, since one of the $k$ pieces meets an $n$-element subset in at least $n/k$ elements; and that every Rider set is a finite union of quasi-independent sets, which it calls rather easy to check and refers to [5].

Dependencies

Theorem 2.3 of Pisier's reference [4]; the Sidon property of quasi-independent sets with an absolute bound on the Sidon constant; Pisier's reference [5] for the direction from Sidon to (vii).

Bears on

• Problem 774: by the integer case above, the problem's hypothesis on an infinite set of positive integers is equivalent to its being a Sidon set, and the problem asks exactly the case of sets of positive integers of Pisier's closing question (p. 89): "Is every set satisfying (vii) a finite union of quasi-independent sets?" The theorem does not answer that question.
• Problem 963: context only. For sets of reals, quasi-independence in the discrete group $\mathbb R$ is dissociation in that problem's sense, by the splitting argument above, so the finite subsets of one Sidon set of reals contain dissociated subsets of proportional size. The problem asks about every $n$-element set of reals, and the theorem gives no bound on its $f(n)$.