Problèmes ouverts: Théorie analytique des polynômes et analyse harmonique

Library card.

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Aline Bonami, Szilárd Révész, and Bahman Saffari, organizers, "Problèmes ouverts: Théorie analytique des polynômes et analyse harmonique," Institut Henri Poincaré working group 2006--2007, open-problem session of 28 June 2007.

What the research consumes

The card carries the digest of the whole proceedings-style collection. Only one contribution bears on E0774: Myriam Déchamps, "Quelques questions ouvertes sur les racines de l'unité et les ensembles de Sidon" (pp. 18--21). She works in the additive group $\mathbb R^2\cong\mathbb C$, calls a set quasi-independent when it admits no nontrivial finite relation with coefficients in $\{-1,0,1\}$ (dissociated, in E0774's terms), lets $\psi(n)$ be the largest size of a quasi-independent subset of the $n$-th roots of unity, and poses Questions 35--39: whether $\psi(n)/\varphi(n)$ is bounded (35), whether $n/\psi(n)$ is bounded (36), the value of $\psi(n)$ (37), whether the set $e^{2\pi i\mathbb Q}$ of all roots of unity is a finite union of quasi-independent sets (38), and whether that set is Sidon (39). She recalls Pisier's characterization and records the Graham–Ramsey structure theorem as Theorem A with arithmetic properties of $\psi$ as Theorems B--C.

Questions 38 and 39 are the two sides of E0774 on this concrete torsion-free additive set: 39 asks for the proportional-dissociation hypothesis and 38 for the finite-union conclusion, and because quasi-independence is hereditary a finite cover can be made a partition. Boundedness in Question 35 would force negative answers to 38 and 39; boundedness in 36 would support positive ones.

Dates

The collection reports Questions 35--39 as open at the 2007 session and supplies historical evidence for that status only. The dated assessment of Problem 774 records the integer problem as open; the bounded-torsion decomposition theorem of Lewko does not settle the torsion-free setting. Neither status should be read into the other.