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?



Are there recursive sets $X$ with Property A that contain infinitely many incompressible strings?
I assume that $X$ is a set of binary strings and the binary strings are coded by natural numbers in this way: consider this enumeration of all strings: 


$\lambda$, 0,1,00,01,10,11,000,.....


(in which, at first we have the string with length zero, then all strings of length one in alphabetical orde and so on).Now code the $n$ th member of this enumeration by the number $n$.


Now we define the set $X$ as $\{w~:~ |w|=2^i~ for ~some ~i\in \mathbb{N}\}$ ($|w|$ denotes the length of the string $w$).


It can easily checked that $X$ satisfies the property $\textsf{A}$. It is also well-known that we have incompressible strings of any length (because the number 
of Turing machines of length<$n$ is at most $2^n-1$ but we have $2^n$ string of length $n$). Therefore $X$ contains infinitely many incompressible strings. 


