  This paper is devoted to the study of Sidon sets, $\Lambda(p)$-sets and some
related notions for compact quantum groups. We establish several different
characterizations of Sidon sets, and in particular prove that any Sidon set in
a discrete group is a strong Sidon set in the sense of Picardello. We give
several relations between Sidon sets, $\Lambda(p)$-sets and lacunarities for
$L^p$-Fourier multipliers, generalizing a previous work by Blendek and
Michali\u{c}ek. We also prove the existence of $\Lambda(p)$-sets for orthogonal
systems in noncommutative $L^p$-spaces, and deduce the corresponding properties
for compact quantum groups. Central Sidon sets are also discussed, and it turns
out that the compact quantum groups with the same fusion rules and the same
dimension functions have identical central Sidon sets. Several examples are
also included.
