  In the current work, we study the eigenvalue distribution results of a class
of non-normal matrix-sequences which may be viewed as a low rank perturbation,
depending on a parameter $\beta>1$, of the basic Toeplitz matrix-sequence
$\{T_n(e^{\mathbf{i}\theta})\}_{n\in\mathbb{N}}$, $\mathbf{i}^2=-1$. The latter
of which has obviously all eigenvalues equal to zero for any matrix order $n$,
while for the matrix-sequence under consideration we will show a strong
clustering on the complex unit circle. A detailed discussion on the outliers is
also provided. The problem appears mathematically innocent, but it is indeed
quite challenging since all the classical machinery for deducing the eigenvalue
clustering does not cover the considered case. In the derivations, we resort to
a trick used for the spectral analysis of the Google matrix plus several tools
from complex analysis. We only mention that the problem is not an academic
curiosity and in fact stems from problems in dynamical systems and number
theory. Additionally, we also provide numerical experiments in high precision,
a distribution analysis in the Weyl sense concerning both eigenvalues and
singular values is given, and more results are sketched for the limit case of
$\beta=1$
