We call A weakly low for K if there is a c such that for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.
"The K-Degrees, Low for K Degrees,and Weakly Low for K Sets." Notre Dame J. Formal Logic 50 (4) 381 - 391, 2009. https://doi.org/10.1215/00294527-2009-017