A Computational Approach to the Borwein-Ditor Theorem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

Abstract

Borwein and Ditor (Canadian Math. Bulletin 21 (4), 497–498, 1978) proved the following. Let \(\mathcal {A}\subset {\mathbb {R}}\) be a measurable set of positive measure and let \({\left\langle {r_m}\right\rangle }_{m\in \omega }\) be a null sequence of real numbers. For almost all \(z \in \mathcal {A}\), there is m such that \(z+r_m\in \mathcal {A}\).

In this note we mainly consider the case that \(\mathcal {A} \) is \(\varPi ^0_{1}\) and the null sequence \({\left\langle {r_m}\right\rangle }_{m\in \omega }\) is computable. We show that in this case every Oberwolfach random real \(z \in \mathcal {A}\) satisfies the conclusion of the theorem. We extend the result to finitely many null sequences. The conclusion is now that for almost every \(z \in \mathcal {A}\), the same m works for each null sequence.

We indicate how this result could separate Oberwolfach randomness from density randomness.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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