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 | |
| State | Published - Jan 1 2018 |
Fingerprint
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)
Cite this
}
Dimension 1 sequences are close to randoms. / Greenberg, Noam; Miller, Joseph S.; Shen, Alexander; Westrick, Linda Brown.
In: Theoretical Computer Science, Vol. 705, 01.01.2018, p. 99-112.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 -