  We investigate Gevrey asymptotics for solutions to nonlinear parameter
depending Cauchy problems with $2\pi$-periodic coefficients, for initial data
living in a space of quasiperiodic functions. By means of the Borel-Laplace
summation procedure, we construct sectorial holomorphic solutions which are
shown to share the same formal power series as asymptotic expansion in the
perturbation parameter. We observe a small divisor phenomenon which emerges
from the quasiperiodic nature of the solutions space and which is the origin of
the Gevrey type divergence of this formal series. Our result rests on the
classical Ramis-Sibuya theorem which asks to prove that the difference of any
two neighboring constructed solutions satisfies some exponential decay. This is
done by an asymptotic study of a Dirichlet-like series whose exponents are
positive real numbers which accumulate to the origin.
