TITLE: Are there recursive sets $X$ with Property A that contain infinitely many incompressible strings?
QUESTION [6 upvotes]: Let us say a set $X$ satisfies Property A if$$\liminf_{n \to \infty} {{\left|X^{\le n}\right|}\over n} = 0.$$Are there recursive sets $X$ satisfying Property A that contain infinitely many incompressible strings?

REPLY [3 votes]: I'm assuming you call `incompressible' integers whose binary representation $x$ satisfy $K(x)>|x|-c$ for a fixed $c$? 
If that's the case, the answer is no. If $X$ is a computable set of integers and some integer $k$ of representation $x$, with $|x|=N$, belongs to $X$, then $K(x|N) \leq \log |X^{\leq 2^{N+1}}| + c'$ for a fixed $c'$. (Indeed one can describe $x$ by just giving its position as a string of length $N$ inside $X$). But by your assumption on $X$, $\log |X^{\leq 2^{N+1}}| - N$ tends to $-\infty$, so almost all strings in $X$ satisfy, say, $K(x\mid|x|) < |x| - 3c$. You can then remove the the condition $|x|$ and get $K(x) < |x| - 2c$ by a usual padding technique (see the book by Li and Vitanyi, I can elaborate if this is important).