  Motivated by the random covering problem and the study of Dirichlet uniform
approximable numbers, we investigate the uniform random covering problem.
Precisely, consider an i.i.d. sequence $\omega=(\omega_n)_{n\geq 1}$ uniformly
distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of
positive real numbers with limit $0$. We investigate the size of the random set
\[
  \mathcal U (\omega):=\{y\in \mathbb{T}: \ \forall N\gg 1,
  \ \exists n \leq N, \ \text{s.t.} \ \| \omega_n -y \| < r_N \}. \] Some
sufficient conditions for $\mathcal U(\omega)$ to be almost surely the whole
space, of full Lebesgue measure, or countable, are given. In the case that
$\mathcal U(\omega)$ is a Lebesgue null measure set, we provide some
estimations for the upper and lower bounds of Hausdorff dimension.
