Do countries have limitations on how high in the air they can claim?

Say you were flying 50 miles over the US (I know that's not possible in a plane but just for the sake of the hypothetical) would they be able to say something to you about it? That'd be towards the top of the mesosphere, I can't imagine they'd be able to do anything about that
Do countries have limitations on how high in the air they can claim?
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