Every 2-random real is Kolmogorov random

Abstract

We study reals with infinitely many incompressible prefixes. Call A∈ 2^ω Kolmogorov random if (∃^∞ n) C(A↾ n)> n-(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf, Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random. Together with the converse—proved by Nies, Stephan and Terwijn [NST]—this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization of 2-randomness.