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2011•Bulletin of the London Mathematical SocietyRequires access

Diagonally non-recursive functions and effective Hausdorff dimension

Noam Greenberg, Joseph S. Miller

Open publisher page 28 citations

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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What this paper is about

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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OpenAlex reports 28 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available 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.

Key concepts: Mathematics, Hausdorff dimension, Dimension (graph theory), Effective dimension, Pure mathematics, Diagonal, Hausdorff space, Geometry

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