Arithmetic characterizations of Sidon sets

Library card.

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Gilles Pisier, "Arithmetic characterizations of Sidon sets," Bulletin of the American Mathematical Society (New Series) 8 (1983), no. 1, 87--89.

What the research consumes

The card carries the digest: the definitions, Theorems 1 and 2, and the exact limits of the announcement. The research consumes Theorem 2 (p. 89; the article's third page) at full strength: a set $\Lambda\subset\widehat G$ is Sidon exactly when some integer $k$ makes every finite $A\subset\Lambda$ contain a quasi-independent $B\subset A$ with $|B|\geq|A|/k$. For subsets of $\mathbb N\subset\mathbb Z=\widehat{\mathbb T}$, quasi-independence is dissociation, so "proportionately dissociated" is "Sidon," and Pisier's reformulation of the finite-union question is exactly E0774.

Reading depth is claims checked: the statement of Theorem 2 was read clause by clause in the published article, which the library does not hold. Its proof is not in the announcement.

Exact limits

The announcement sends the proofs of Theorem 1, the proposition, and the difficult direction of Theorem 2 to a reference then forthcoming; it gives no dependence of $k$ on the Sidon constant, so the extraction constants obtained through it are positive but not numerically quantified; and it answers neither the finite-union question nor supplies a block construction.