Some new thin sets of integers in Harmonic Analysis

Details Some new thin sets of integers in Harmonic Analysis Some new thin sets of integers in Harmonic Analysis

2009-12-21

arxiv.org/abs/0912.4214v1

Journal d'Analyse Math\'ematique 86 (2002) 105-138

Mathematics

Functional Analysis

Scientific paper

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.

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