### Abstract

Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

Original language | English (US) |
---|---|

Pages (from-to) | 789-811 |

Number of pages | 23 |

Journal | Mathematical Structures in Computer Science |

Volume | 16 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)
- Computer Science Applications

### Cite this

*Mathematical Structures in Computer Science*,

*16*(5), 789-811. https://doi.org/10.1017/S0960129506005469

}

*Mathematical Structures in Computer Science*, vol. 16, no. 5, pp. 789-811. https://doi.org/10.1017/S0960129506005469

**Schnorr dimension.** / Downey, Rodney; Merkle, Wolfgang; Reimann, Jan Severin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Schnorr dimension

AU - Downey, Rodney

AU - Merkle, Wolfgang

AU - Reimann, Jan Severin

PY - 2006/10/1

Y1 - 2006/10/1

N2 - Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

AB - Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

UR - http://www.scopus.com/inward/record.url?scp=33750738235&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750738235&partnerID=8YFLogxK

U2 - 10.1017/S0960129506005469

DO - 10.1017/S0960129506005469

M3 - Article

VL - 16

SP - 789

EP - 811

JO - Mathematical Structures in Computer Science

JF - Mathematical Structures in Computer Science

SN - 0960-1295

IS - 5

ER -