I am looking to better understand the structure of the set $E_\alpha$ associated to the Jarnik's theorem (according to Wolff T.H, Lectures on Harmonic Analysis, Ch. 9, pp̣. 67): Let $\alpha > 0$ be fixed and define
$$E_\alpha=\left\{x\in\mathbb{R}\mid \text{exists infinitely many unique rationals $\frac{p}{q}$ such that }\left|x - \frac{p}{q}\right|\leq q^{-(2 + \alpha)}\right\}$$
Then what seems to be the Jarnik's Theorem is that the set $E_\alpha$ has a Hausdorff dimension of $\frac{2}{2 + \alpha}$. I am not that interested in the theorem itself, but rather on the structure of the set $E_\alpha$. Specifically, what would be some examples of sets $E_\alpha$? Do we know any qualitative properties of the elements of $E_\alpha$? Clearly $E_\alpha\neq \varnothing, \forall \alpha > 0$ for $\forall \alpha > 0:0\in E_\alpha$. Do we know what kinds of elements are included to $E_\alpha$ from a small (or large) neighbourhood of the origin?
If you happen to know a good source discussing these things, feel free to post it!