Dirichlet's theorem says that all numbers $x\in [0,1]$ can be approximated by infinitely many fractions $p/q \in \mathbb{Q}$ with error $|x - p/q| \le 1/q^2$.

I am interested in the following question: Fix $m>0$. Let $S(m)$ denote the set of $x\in [0,1]$ for which no fraction $p/q \in \mathbb{Q}$ with $q<100\sqrt{m}$ approximates $x$ with error smaller than $1/m$, i.e., $|x-p/q| > 1/m$ for any such choice of $p,q$. What bounds are there for the measure of $S(m)$?

EDIT: Let $\Phi(q)$ denote the set of numbers smaller than $q$ coprime to $q$. And let $$I(q) = \bigcup_{p\in \Phi(q)} [p/q, p/q+1/m]$$ For $q_1,q_2<\sqrt{m}$ the sets $I(q_1),I(q_2)$ will be disjoint. This can give a simple lower bound (by analyzing $\sum_{k<\sqrt{m}} \phi(k)$, which is a well known problem). I'm interested in if anything stronger than this bound can be given.