The dichotomy problem for Sidon sets and sets of analyticity
For $\Lambda \subset \mathbb{Z}$ let $A(\Lambda) = \{(\hat f(\lambda))_{\lambda \in \Lambda} : f \in L^1(\mathbb{T})\}$. Sidon sets are meant in the sense of harmonic analysis: $\Lambda$ is Sidon if and only if $A(\Lambda) = c_0(\Lambda)$. $\Lambda$ is a set of analyticity if every function operating on $A(\Lambda)$ is analytic. Kahane and Katznelson showed that a random set $\Lambda$ with $\mathbb{P}(n \in \Lambda) = p_n$ is almost surely Sidon when $n p_n$ stays bounded, and almost surely a set of analyticity when $n p_n \to \infty$.
Problem. Is each $\Lambda \subset \mathbb{Z}$ a Sidon set or a set of analyticity?
Source: Raised in commutative harmonic analysis in the 1960s; see J.-P. Kahane and Y. Katznelson. Listed as Problem 96 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- FA
- Keywords
- commutative harmonic analysis, Fourier algebra, functions operating on Fourier coefficients, sets of analyticity, Sidon sets
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{katznelson1966malliavin,
author = {Katznelson, Yitzhak and Malliavin, Paul},
title = {V{\'e}rification statistique de la conjecture de la dichotomie sur une classe d'alg{\`e}bres de restriction},
journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences, S{\'e}rie A-B},
volume = {262},
pages = {A490--A492},
year = {1966}
}
@article{kahane2007katznelson,
author = {Kahane, Jean-Pierre and Katznelson, Yitzhak},
title = {Entiers al{\'e}atoires, ensembles de {S}idon, densit{\'e} dans le groupe de {B}ohr et ensembles d'analyticit{\'e}},
journal = {Comptes Rendus Math{\'e}matique},
volume = {345},
number = {1},
pages = {21--24},
year = {2007},
doi = {10.1016/j.crma.2007.06.001},
eprint = {0705.4261},
archivePrefix = {arXiv}
}
@article{kahane2008katznelson,
author = {Kahane, Jean-Pierre and Katznelson, Yitzhak},
title = {Entiers al{\'e}atoires et analyse harmonique},
journal = {Journal d'Analyse Math{\'e}matique},
volume = {105},
pages = {363--378},
year = {2008},
doi = {10.1007/s11854-008-0040-6}
}
@article{pisier1983,
author = {Pisier, Gilles},
title = {Arithmetic characterizations of {S}idon sets},
journal = {Bulletin of the American Mathematical Society (N.S.)},
volume = {8},
number = {1},
pages = {87--89},
year = {1983},
doi = {10.1090/S0273-0979-1983-15092-9}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 96; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00959, "The dichotomy problem for Sidon sets and sets of analyticity", clopen.solutions.