Sidon sets and Riesz products

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Jean Bourgain, Sidon sets and Riesz products, Annales de l'Institut Fourier 35, no. 1 (1985), 137–148, DOI:10.5802/aif.1003. The selected text is the NUMDAM digitization, with an archive cover followed by the twelve journal pages. The journal's article record confirms this identity.

The PDF cover prints “© Annales de l'institut Fourier, 1985, tous droits réservés.” The notice and journal article record were read 2026-10-15; no open redistribution license was found on that record. No file of this source is held.

Scope

Here Sidon means the harmonic-analysis property for a subset $\Lambda$ of the dual of a compact abelian group: the sum of absolute coefficients is bounded by a constant times the supremum norm of the associated character polynomial. It is not merely the additive $B_2$ condition on pair sums.

The theorem in Section 2, printed pp. 138–139, includes the following equivalent condition. There is $\delta>0$ such that every finitely supported scalar family on $\Lambda$ has a quasi-independent subset retaining at least $\delta$ times its total absolute weight. Quasi-independence forbids nonzero finite relations with coefficients in $\{-1,0,1\}$. The same theorem includes the unweighted proportional-subset condition.

Read status: claims checked. The definitions and Section 2 theorem were read at statement depth, with the theorem checked against the images of pp. 138–139. Its proof and quantitative dependence of the extraction constant have not been checked here.

Bears on. Problem 774: the weighted characterization is a prospective input for arguments involving proportional dissociation. It does not itself establish the requested finite partition into dissociated subsets of the positive integers.