I am sorry if this is a dump question but I can't find any reference for this. What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$?
I speculate this is $1$ but don't know how to prove it rigorously.
2026-03-30 06:32:43.1774852363
What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$
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$d$-dimensional Hausdorff measure for any $d > 1$ is nonatomic, so any countable set has $d$-dimensional Hausdorff measure $0$. In particular the rationals have $1$-dimensional Hausdorff measure $0$, and the irrationals have infinite $1$-dimensional Hausdorff measure. On the other hand the $d$-dimensional Hausdorff measure of $\mathbb R$ is $0$ for $d > 1$. So the irrationals have Hausdorff dimension $1$.