  We consider the one-dimensional NLS equation with a convolution potential and
a quintic nonlinearity. We prove that, for most choices of potentials with
polynomially decreasing Fourier coefficients, there exist almost-periodic
solutions in the Gevrey class with frequency satisfying a Bryuno non-resonance
condition. This allows convolution potentials of class $C^p$, for any integer
$p$: as far as we know this is the first result where the regularity of the
potential is arbitrarily large and not compensated by a corresponding smoothing
of the nonlinearity.
