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Diagonally non-recursive functions and effective Hausdorff dimension

Diagonally non-recursive functions and effective Hausdorff dimension We prove that every sufficiently slow-growing diagonally non-recursive (DNR) function computes a real with effective Hausdorff dimension 1. We then show that, for any recursive unbounded and non-decreasing function j , there is a DNR function bounded by j that does not compute a Martin-Löf random real. Hence, there is a real of effective Hausdorff dimension 1 that does not compute a Martin-Löf random real. This answers a question of Reimann and Terwijn. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Diagonally non-recursive functions and effective Hausdorff dimension

Diagonally non-recursive functions and effective Hausdorff dimension

Bulletin of the London Mathematical Society , Volume 43 (4) – Aug 1, 2011

Abstract

We prove that every sufficiently slow-growing diagonally non-recursive (DNR) function computes a real with effective Hausdorff dimension 1. We then show that, for any recursive unbounded and non-decreasing function j , there is a DNR function bounded by j that does not compute a Martin-Löf random real. Hence, there is a real of effective Hausdorff dimension 1 that does not compute a Martin-Löf random real. This answers a question of Reimann and Terwijn.

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References (17)

Publisher
Oxford University Press
Copyright
© 2011 London Mathematical Society
Subject
PAPERS
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdr003
Publisher site
See Article on Publisher Site

Abstract

We prove that every sufficiently slow-growing diagonally non-recursive (DNR) function computes a real with effective Hausdorff dimension 1. We then show that, for any recursive unbounded and non-decreasing function j , there is a DNR function bounded by j that does not compute a Martin-Löf random real. Hence, there is a real of effective Hausdorff dimension 1 that does not compute a Martin-Löf random real. This answers a question of Reimann and Terwijn.

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Aug 1, 2011

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