A Computational Approach to the Borwein-Ditor Theorem

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.