Actually, "how high above the dirt you own" is already well defined, both in law, regulation and court cases. What is happening at the ground and a bit above on your property is yours and you can even own copyright on it. Government and individuals are not supposed to snoop on you. Fliers are not supposed to fly below 500' above you in rural areas, higher in urban areas. The result is that if someone is peeping on you in a situation where you can reasonably expect privacy you can sue them and they can also separately be fined and imprisoned by the government. The government itself is supposed to get a warrant to view you.
Just as importantly, the other logical question is how far down below the surface do you own. This is your mineral rights.
Typically in the eastern United States you own all the way down, let's say to the mantle. The exact depth doesn't get too much precedence since deep drilling hasn't been done but fracking may be changing that soon.
Out in the western United States you may well not own below your surface soil, you may well not own your mineral rights, you may well not own your water rights and you may well not even own the rain that falls upon your land. Check your deed and your state laws.
In the fourth dimension of space you own nothing. That is possibly reserved for the alternative owners who may or may not own the land you own in this variation of the multiverse. This will not appear on your deed.
In the fifth dimension (time) your ownership started at some point in the past delineated by your purchase and will end at some point in the future marked by your death or other event. Again, check your deed for the particulars.