Some new thin sets of integers in Harmonic Analysis

Daniel Li (1) , Hervé Queffélec (2) , Luis Rodriguez-Piazza (3)

1 LML - Laboratoire de Mathématiques de Lens
2 LPP - Laboratoire Paul Painlevé
3 Departamento de Analisis Matematico

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in $\Lambda$ have uniformly convergent series, and their Fourier coefficients are in $\ell_p$ for all $p>1$; moreover, all the Lebesgue spaces $L^q_\Lambda$ are equal for $q<+\infty$. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in $\Lambda$ is non separable. So these sets are very different from the thin sets of integers previously known.

Keywords: uniformly distributed set; ergodic set; lacunary set; $\Lambda(q)$-set; quasi-independent set; random set; $p$-Rider set; Rosenthal set; $p$-Sidon set; set of uniform convergence; uniformly distributed set.

Domains: Functional Analysis [math.FA]

Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. Some new thin sets of integers in Harmonic Analysis. Journal d'analyse mathématique, 2002, 86, pp.105-138. ⟨hal-00442352⟩