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Let $\mu (r)>2$ be the irrationality measure of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$:

$P=\{r\in \mathbb{R}: \mu(r)=Constant\}$

Is this set a fractal, and If so, then what is it's dimension?

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    Perhaps to answer (1) I should try and motivate my question a bit more: i just learned about irrationality measure last week, and I think it's really cool. It's called a "measure", and the most basic geometrical shape is the circle, so after thinking about these $P$ sets, and not finding any references online, it seemed like $P$ must have a fractal structure. Also, as soon as I see power law equations with non-integer exponents, the first thing I think of is fractals. It seemed like there might be a nice proof that this set is fractal and maybe even a nice way to compute its dimension.2010-11-19

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It is a fractal much like the cantor set with dimension 2/r. That is Jarniks theorem. You can find a proof in the Falconer book Fractal Geometry: Mathematical Foundations and Applications.

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    Thanks! I own this book, I should have checked it more thoroughly.2011-02-19