Dimension 1 sequences are close to randoms

TL;DR

This paper characterizes sequences with effective Hausdorff dimension 1 as those coarsely similar to Martin-Löf random sequences, and extends the analysis to sequences of arbitrary effective dimension, providing optimal bounds for dimension elevation through minimal bit modifications.

Contribution

It establishes a precise equivalence between effective Hausdorff dimension and coarse similarity to random sequences, and determines optimal bounds for increasing a sequence's dimension with minimal changes.

Findings

Sequences of dimension 1 are coarsely similar to Martin-Löf randoms.

Sequences of arbitrary dimension s are coarsely similar to weakly s-random sequences.

Minimal bit modifications can increase a sequence's dimension to t, with the bound being optimal.

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.