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?
2026-03-28 21:27:03.1774733223
Sets of Constant Irrationality Measure
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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.