>I see that The Masked Avenger has already got an answer (21h ago at the time of this writing :-). Since I also got my counterexample(s) let me state it here. At least I think that my presentation is perhaps more direct.

Let *harp* $\ [a;b),\ $ where $\ 0<a<b\ $ are positive reals, be the set of reals $\ t\ $ such that $\ a<t<b,\ $ and $\ \frac ta\ $ is an integer (i.e. $\ a\,|\,t;\ $ the musical name ***harp*** stands for ***homogenous arithmetic progressions***).

Let $\ n>2\ $ be an even integer. Consider the following two harps:

$$A\ :=\ [n-1;\, 2\!\cdot\!(n^2-1))$$
$$B\ :=\ [n+1;\, 2\!\cdot\!(n^2-1))$$

Also, let

$$\ C\ :=\ A\cup B$$

Let $\ x:=2,\ $ and

$$\ d_H\ :=\ \frac {\left|\,\{y\in H: x|y\}\,\right|}{\left|H\right|}$$

for arbitrary harp $\ H.\ $ Then:

$$d_A\ :=\ \frac n{2\cdot n+1}$$
$$d_B\ :=\ \frac{n-2}{2\cdot n -3}$$
$$d_C\ :=\ \frac{2\cdot n - 2}{4\cdot n-3}$$

and

$$d_B\ <\ d_A\ <\ d_C$$

for every even integer $\ n>2.$