  This short and simple communication is motivated by recent papers by L.
Colzani and A. Kochergin. We give a brief analysis of an example by
Poincar\'{e} related to sums of the type $ \sum_{k=0}^{t-1} f(k{\alpha}+{x})$
where $f$ is a continuous periodic function and $\alpha$ is irrationaland its
recent generalisations. Most of the constructions under consideration are
well-known. In this note, we just wanted to bring all the results together and
give a general and improved multi-dimensional formulation of a recent result by
A. Kochergin, prove non-existence of a universal continuous function and
discuss some of the related results in terms of Diophantine Approximation. In
particular, in our opinion smoothness results involving Diophantine exponents
$\omega$, $\hat{\omega}$ and $\hat{\lambda}$ had never been documented before.
