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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Research supported by the Marsden fund of New Zealand and the Lion foundation of New Zealand.
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Galicki, A., Nies, A. (2016). A Computational Approach to the Borwein-Ditor Theorem. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_10
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DOI: https://doi.org/10.1007/978-3-319-40189-8_10
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