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Compressibility of Infinite Binary Sequences

It is known that infinite binary sequences of constant 
Kolmogorov complexity are exactly the recursive ones. 
Such a kind of statement no longer holds in the presence of resource bounds. 
Contrary to what intuition might suggest, there are sequences of 
constant, polynomial-time bounded Kolmogorov complexity that are 
not polynomial-time computable. This motivates the study of 
several resource-bounded variants in search for a characterization, 
similar in spirit, of the polynomial-time computable sequences. 
We propose some definitions, based on Kobayashi's notion of 
compressibility, and compare them to both the standard resource-bounded 
Kolmogorov complexity of infinite strings, and the uniform complexity. 
Some nontrivial coincidences and disagreements are proved. 
The resource-unbounded case is also considered.