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Does there exist $x>2$ such that uncountably many reals have irrationality measure x? Must there exist at least one number of irrationality measure $x$?

related question on sets of constant irrationality measure

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the question you linked contains an answer: by Jarnik-Besikovitch theorem, the set of real numbers of any given irrationality measure has non-zero Haussdorff dimension, so it is uncountable (hence non-empty) –  user8268 Jul 6 '12 at 8:47

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