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We show that a sequence has effective Hausdorff dimension 1 if and only if itis coarsely similar to a Martin-L\"{o}f random sequence. More generally, asequence has effective dimension $s$ if and only if it is coarsely similar to aweakly $s$-random sequence. Further, for any $s<t$, every sequence of effectivedimension $s$ can be changed on density at most $H^{-1}(t)-H^{-1}(s)$ of itsbits to produce a sequence of effective dimension $t$, and this bound isoptimal.

Dimension 1 sequences are close to randoms