Martin-Löf randomness and Galton–Watson processes

The members of Martin-Löf random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Löf random Galton–Watson trees with survival parameter 23. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than γ=log232, and a necessary condition is to have effective Hausdorff dimension greater than or equal to γγ.

► We show that our distribution is effectively equivalent to that of Cenzer et al.
► We study reals belonging to Martin-Löf random closed sets under such a distribution.
► We show that these reals all have high effective Hausdorff dimension.
► Conversely, they include all reals of sufficiently high effective Hausdorff dimension.