  We show that Ambro-Kawamata's non-vanishing conjecture holds true for a
quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an
ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further
show that the general element of |H| is smooth. We then verify Ambro-Kawamata's
conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita's
freeness conjecture for a Gorenstein quasi-smooth weighted hypersurface.
  For the proofs, we introduce the arithmetic notion of regular pairs and
enlighten some interesting connection with the Frobenius coin problem.
