  Thurston boundary of the universal Teichm\"uller space $T(\mathbb{D})$ is the
space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of
$\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of generalized Teichm\"uller
type if it shrinks the vertical foliation of a holomorphic quadratic
differential. We provide the first examples of generalized Teichm\"uller rays
which diverge near Thurston boundary $PLM_{bdd}(\mathbb{D})$. Moreover, for
every $k\geq 1$ we construct examples of rays with limit sets homeomorphic to
$k$-dimensional cubes. For the latter result we utilize the classical Kronecker
approximation theorem from number theory which states that if
$\theta_1,\ldots,\theta_k$ are rationally independent reals then the sequence
$(\{\theta_1 n\},\ldots,\{\theta_k n\})$ is dense in the $k$-torus
$\mathbb{T}^k$.
