Average case problem and worst case problem in lattice
According to my understanding, in cryptographic constructions based on worst-case hardness we based the cryptography scheme on hard subsets in which the problem is hard, so we should also avoid we based the cryptography scheme on subsets in which the problem is easy.


I think the idea of the statement is more along the following line:

"For most cryptographic schemes we have no proof that the distribution we use in practice is actually as hard to solve as the hardest ones we could come up with. With lattices however we have a proof that the sampling strategy we use generates problems that are just as hard to solve as the hardest instances we could come up with."

So the statement is about the average case for lattices being provably as difficult as the worst-case.