A non-vanishing result for weighted complete intersections

Let $X$ be a smooth (or mildly singular) projective variety and let $H$ be an ample line bundle on $X$. Kawamata conjectured that if $H-K_X$ is ample, then the linear system $|H|$ is not empty. I will explain that the conjecture holds true for weighted complete intersections which are Fano or Calabi-Yau, relating it with the Frobenius coin problem.
This is based on a joint work with M. Pizzato and T. Sano.