📝 SummarXiv

Abstract

A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we investigate the Kolmogorov complexity and the binary expansions of a very specific subset of the left-computable numbers. We show in our main result that a real number is reordered computable if, and only if, it is left-computable and not strongly Kurtz random. In preparation of this, we characterize strong Kurtz randomness by a suitable notion of randomness tests. Recently, in a contribution to the conference CiE 2025, the authors looked at the binary expansions of reordered computable numbers and clarified whether they can be immune or hyperimmune. We take this topic up again and extend our analysis to the notions hyperhyperimmune, strongly hyperhyperimmune, and cohesive. It turns out that by using our main theorem and a number of well-known facts we can obtain a complete picture of the situation, obtaining shorter proofs of results in the conference paper and new results. Then, we investigate the effective Hausdorff and packing dimensions of reordered computable numbers. Finally, we have a short look at regular reals in the context of immunity properties, Kolmogorov complexity and (strong) Kurtz randomness.