Dimension 1 sequences are close to randoms

Noam Greenberg; Joseph S. Miller; Alexander Shen; Linda Brown Westrick

doi:10.1016/j.tcs.2017.09.031

Dimension 1 sequences are close to randoms

Noam Greenberg, Joseph S. Miller, Alexander Shen, Linda Brown Westrick

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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)
Pages (from-to)	99-112
Number of pages	14
Journal	Theoretical Computer Science
Volume	705
DOIs	https://doi.org/10.1016/j.tcs.2017.09.031
State	Published - Jan 1 2018

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

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10.1016/j.tcs.2017.09.031

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

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AU - Greenberg, Noam

AU - Miller, Joseph S.

AU - Shen, Alexander

AU - Westrick, Linda Brown

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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.

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VL - 705

SP - 99

EP - 112

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -

Dimension 1 sequences are close to randoms

Abstract

All Science Journal Classification (ASJC) codes

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