If $t$ is a time constructible function then we can define the time-bounded Kolmogorov complexity of a string $x$ as:
$C^t(x)$ = the size of the smallest program p that generates $x$ in $t(|x|)$ steps
We can also extend the incompressibility notion: a string $x$ is incompressible if $C^t(x) \ge |x|$ (otherwise $x$ is compressible).
Now we can define the function:
$f^t(n) = | \{ x \text{ : } |x|=n \text{ and } x \text{ is compressible with respect to } C^t \} |$
( $f^t(n)$ is the total number of compressible strings of length n ).
What is the relation (if any) between $t$ and the rate of growth of $f$? Can we say something about it?
EDIT: For example, can we choose $t$ to make $f$ grow polynomially? ... NO