fallibilism and knowledge

It is obvious that knowledge should not require that the knowing subject be infallible in getting at the true belief. So infallibilism is false; knowledge should be compatible with fallibilism. In other words, at least in some cases, a bit of luck or grace is involved in the process of the knower's reaching the truth. But does this imply that Gettier-style problems are unavoidable?

Gettier's two initial cases obviously involve luck; so does the Nogot-havit case; the fake barn case, too; and also the Jill-assassination case and the Tim-Tom case. So all the Gettier-style cases I can think of involve getting at some true belief by luck. While on the other hand, we've already claimed that knowledge is compatible with the existence of luck. So, one question to ask is, what kind of luck is allowed and what kind of luck is not? There is no need to survey the long list of attempts at coming up with new or novel epistemic norms for knowledge to solve the Gettier problems here. A rough impression I get frm looking at them is that pretty much all that they do is distinguish the luck that is allowed from that which is not. But if all these attempts exceptionlessly fail, then what conclusion should we draw? One option is to remain optimistic and look at the future brightly. The other option is to admit that there is simply no such universal formula for us to distinguish permissible luck from unpermissible luck, and hence claim that fallibilism is unacceptable. That will leave us in a dilemmatic quandary.