  We introduce a general notion of "genericity" for countable subsets of a
space with Borel measure, and apply it to the set of vertices in the curve
complex of a surface S, interpreted as subset of the space of projective
measured laminations in S, equipped with its natural Lebesgue measure.
  We prove that, for any 3-manifold M, the set of curves c on a Heegaard
surface S in M, such that every non-trivial Dehn twist at c yields a Heegaard
splitting of high distance, is generic in the set of all essential simple
closed curves on S.
  Our definition of "genericity" is different and more intrinsic than
alternative such existing notions, given e.g. via random walks or via limits of
quotients of finite sets.
