Let $\ f(m)\ $ be the number of the $m$-th partition intervals which intersect $\ A.\ $
Theorem 1 There exists $\ A\ $ which is not dense in any non-trivial interval whenever
$$ \lim_{m=\infty} \frac {f(m)}m\ =\ 0 $$
Proof The required $\ A\ $ can be given as follows:
$$ A\ :=\\ \ \left\{ \frac {2\cdot k-1}{2\cdot m}: \ m\in\Bbb N\ \ \text{and} \ \ k\in\Bbb Z\ \ \text{and}\ \ 0<k\le f(m)\,\right\} $$
End of Proof
On the other hand,
Theorem 2 If $\ \lim_{m=\infty} f(m) = \infty\ $ then there exists respective $\ A\ $ which is dense in $\ [0;1]$.
Proof The required $\ A\ $ can be given by:
$$ A\ :=\\ \ \left\{ \frac{2\cdot k-1}{2\cdot f(m)}:\ m\in\Bbb N \ \ \text{and}\ \ f(m)>0\ \ \text{and}\ \ k\in\Bbb Z \ \ \text{and}\ \ 0<k\le f(m) \right\} $$
End of Proof