Diagonally non-recursive functions and effective Hausdorff dimension
Abstract
We prove that every sufficiently slow-growing diagonally non-recursive (DNR) function computes a real with effective Hausdorff dimension 1. We then show that, for any recursive unbounded and non-decreasing function j , there is a DNR function bounded by j that does not compute a Martin-Löf random real. Hence, there is a real of effective Hausdorff dimension 1 that does not compute a Martin-Löf random real. This answers a question of Reimann and Terwijn.