  The covariant gauges are known to suffer from the Gribov problem: even after
fixing a gauge non-perturbatively, there may still exist residual copies which
are physically equivalent to each other, called Gribov copies. While the
influence of Gribov copies in the relevant quantities such as gluon propagators
has been heavily debated in recent studies, the significance of the role they
play in the Faddeev--Popov procedure is hardly doubted. We concentrate on
Gribov copies in the first Gribov region, i.e., the space of Gribov copies at
which the Faddeev--Popov operator is strictly positive (semi)definite. We
investigate compact U($1$) as the prototypical model of the more complicated
standard model group SU($N_{c}$). With our Graphical Processing Unit (GPU)
implementation of the relaxation method we collect up to a few million Gribov
copies per orbit. We show that the numbers of Gribov copies even in the first
Gribov region increase exponentially in two, three and four dimensions.
Furthermore, we provide strong indication that the number of Gribov copies is
gauge orbit dependent.
