TY - JOUR

T1 - Dimension 1 sequences are close to randoms

AU - Greenberg, Noam

AU - Miller, Joseph S.

AU - Shen, Alexander

AU - Westrick, Linda Brown

N1 - Funding Information:
The first and fourth authors were supported by a Rutherford Discovery Fellowship from the Royal Society of NZ. The second author was partially supported by grant #358043 from the Simons Foundation. The third author was partially supported by RaCAF ANR-15-CE40-0016-01 grant and RBFR 16-01-00362 A.
Publisher Copyright:
© 2017 Elsevier B.V.

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.

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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 -