Example of the real numbers $x$ appearing in the Jarnik's theorem: $||nx||\leq n^{-\beta}$ for infinitely many $n\in\mathbb{N}$ for fixed $\beta>1$

Question

According to Falconer (Falconer, '85, The Geometry of Fractal Sets, theorem 8.16, p. 134), a part of the Jarnik's theorem is the following:

Take $\beta>1$. The set of real numbers $x$ for which the intequality $||nx||\leq n^{-\beta}$ holds for infinitely many $n\in\mathbb{N}$ has Hausdorff dimension $2/(1 + \beta)$.

where $||.||$ is the distance to the nearest integer in regular Euclidean norm.

Falconer doesn't provide an example of such a real number $x$ so I got curious about knowing some examples of $x$s, besides the obvious candidate $x = 0$. Do you know some or do you know a good source for this?