We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-Löf random sequence. More generally, a sequence has effective dimension s if and only if it is coarsely similar to a weakly s-random sequence. Further, for any s<t, every sequence of effective dimension s can be changed on density at most H−1(t)−H−1(s) of its bits to produce a sequence of effective dimension t, and this bound is optimal.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)