Suppose that $A$ is a given subset of $I=[0,1]$.

$\displaystyle \left\{ x_j = \frac{j}{m} \right\}_{j=0}^{m}$ is a partition of $I$. $n$ is the number of $[x_{i-1},x_{i}]$ such that $[x_{i-1},x_{i}]\cap A \neq \emptyset$. If there exists an $M>0$, where $n$ is always greater than $\ln m$ for $\forall m \in \mathbb{N}+$ with $m>M$, is it possible to prove that $A$ is dense on some subinterval of $I$?

A brief line: (1). $A$ is given. 
(2). $M$ can or cannot be found due to $A$ to satisfy "For $\forall m > M$, it always holds $n\geq \ln m$."
(3). If such $M$ can be found, whether can we assure that $A$ is dense on some subinterval of $I$?

Notes: I want to emphasize that you cannot construct $A$ with $m$ related to the precondition, because $m$ is given and changing to $\infty$ after $A$ has been supposed. Even if you want to construct $A$ with some variables, please notice that when $A$ is constructed, your variables are fixed and cannot follow the change of $m$.