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Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it measurable?

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    If $x$ has irrationality measure $\mu$, and $y$ has irrationality measure $\tau<\mu$, does $x+y$ automatically have irrationality measure $\mu$? Linguistically, I'd want this to be true. It seems like it might be helpful.2012-03-16
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    See also http://mathoverflow.net/questions/249194/irrationality-measure-of-the-number-is-itself2016-09-06

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