Suppose that $A$ is a 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 $[x_{i-1},x_{i}]\cap A \neq \emptyset$. When $m\rightarrow \infty$, if $n$ is always greater than $\ln m$, is it possible to prove that $A$ is dense on some subinterval of $I$?