Time-bounded incompressibility of compressible strings and sequences

For every total recursive time bound t, a constant fraction of all compressible (low Kolmogorov complexity) strings is t-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length n is compressible to logn yet t-bounded incompressible below ; and there is a countably infinite number of recursive infinite sequences of which every initial segment is similarly t-bounded incompressible. These results and their proofs are related to, but different from, Barzdins's lemma.