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Dimension 1 sequences are close to randoms (1709.05266v1)
Published 15 Sep 2017 in math.LO, cs.IT, and math.IT
Abstract: We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-L\"{o}f 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.