Each battle where they actually do a hit, so a 0-0 wouldn't count. Should have made that clearer! Also, rule is needed for building. I didn't think to clarify that.
But for the battle with the outcome 3:0, when they deal 3 hits, do they get 1 improvement, or 3 improvements?
from what I understand now, just the one! ;)
I think so also, but wanted to be 100% sure.
A 3-0 would only get 1 improvement. So long as one die is 1 or greater, you get 1 improvement.