  The members of Martin-L\"of random closed sets under a distribution studied
by Barmpalias et al. are exactly the infinite paths through Martin-L\"of random
Galton--Watson trees with survival parameter $\frac{2}{3}$. To be such a
member, a sufficient condition is to have effective Hausdorff dimension
strictly greater than $\gamma=\log_2 \frac{3}{2}$, and a necessary condition is
to have effective Hausdorff dimension greater than or equal to $\gamma$.
