Re: You don't own the air above your ground
As far as actual ownership is concerned, if wikipedia and quora are to be believed then US landowners own rights to all minerals under their property barring an explicit notation in the deed. This apparently even includes the oil that flows into the rock under your property from under your neighbor's property due to the fact that you've just sucked your property dry. You are *not* allowed to dig down and then sideways under someone else's property without a lease, however.
This fits with what I've always heard as far as the US is concerned, in that US law on this differs greatly from European law.