  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.
