It is well known that for almost every $c \in \mathbb{R} / \mathbb{Z}$ there exists $(q_n)_{n \geq 1}$ and $(a_n)_{n \geq 1}$ such that $$|c - a_n / q_n| \leq 1/ q_n^2,$$ where $q_n < q_{n+1} \leq q_n^{1+ \epsilon}$ for every $\epsilon > 0$.
The problem with the above is that we have no idea when the sequence $(q_n)_{n \geq 1}$ starts. That is, how large is $q_1$ (assume that we take $q_1$ to be minimal so that the above holds). Do we know anything about the set of $c$ for which $q_1 \leq T$ for some fixed large $T$? Has this set been studied before?