You can alwats erect a landmark in somebody's territory for a considerable diplomatic boost. With major civs it is something like a +20 boost, with transparent diplomacy on, for the first landmark, and diminishing boosts for the consequent ones. That brings me to my other point.
AI will usually ask you to stop digging as soon as you start digging, not as soon as you finish a dig, then deciding to either take an artifact home or construct a landmark. You can only respond to that demand in two ways: ignore the demand or promise not to dig again. Well, here's the thing: if you promise not to dig, but then you keep doing that while only building landmarks on the AI territory, they don't mind. That doesn't count as breaking the promise, instead it just keeps on giving positive diplo boosts. It seems to me that you only break the promise once you finish digging up a site and then choose to take an artifact (but don't take my word for this).
In my games this means digging everywhere I can, denying the AI digs by constructing landmarks for them - often in bad terrain like desert or outside of city working range - getting positive diplo reputation, all while having promised not to ever dig in their territory again and somehow ending up not breaking the promise.