### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 99-112 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 705 |

DOIs | |

State | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*705*, 99-112. https://doi.org/10.1016/j.tcs.2017.09.031

}

*Theoretical Computer Science*, vol. 705, pp. 99-112. https://doi.org/10.1016/j.tcs.2017.09.031

**Dimension 1 sequences are close to randoms.** / Greenberg, Noam; Miller, Joseph S.; Shen, Alexander; Westrick, Linda Brown.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Dimension 1 sequences are close to randoms

AU - Greenberg, Noam

AU - Miller, Joseph S.

AU - Shen, Alexander

AU - Westrick, Linda Brown

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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−1(t)−H−1(s) of its bits to produce a sequence of effective dimension t, and this bound is optimal.

AB - 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−1(t)−H−1(s) of its bits to produce a sequence of effective dimension t, and this bound is optimal.

UR - http://www.scopus.com/inward/record.url?scp=85033449658&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85033449658&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2017.09.031

DO - 10.1016/j.tcs.2017.09.031

M3 - Article

AN - SCOPUS:85033449658

VL - 705

SP - 99

EP - 112

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -