Thanks to symetry of information (i.e $\forall x,y, K(xy) = K(x) + K(y|x) - O(log(|x| + |y|)$), one can easily show that :

$ \exists N \forall x, (|x| = n^{log(n)} and |x| \geq N), \exists y, (|y| \leq n), K(xy) > K(x) $.

But if we consider the Kolmogorov complexity with bounded ressources (i.e $K^{f(n)}(x)$ is the length of the shortest program that runs in time $f(n)$ and outputs x.

Do we have a similar property, that

$\forall x ,|x| = n^{log(n)}, \exists y, |y| \leq n, K^{2^n}(xy) > K^{2^n}(x) $

?